#499602

order	512
nilpotency class	4
isoclinism	isoclinic class of #499262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3184
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23761
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

460 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,8)(4,18)(5,10)(6,11)(7,12)(9,21)(13,29)(14,30)(15,32)(16,33)(17,31)(19,34)(20,35)(22,51)(23,42)(24,37)(25,38)(26,56)(27,40)(28,41)(36,55)(39,44)(43,57)(45,62)(46,59)(47,58)(48,61)(49,60)(50,54)(52,63)(53,64);(1,3,13,16,5,17,7,28)(2,8,29,33,10,31,12,41)(4,19,20,53,21,54,23,48)(6,25,43,49,24,52,27,15)(9,34,42,64,18,50,35,61)(11,38,57,60,37,63,40,32)(14,44,47,22,46,26,45,55)(30,56,58,36,59,39,62,51);(1,4)(2,9)(3,14)(5,21)(6,26)(7,20)(8,30)(10,18)(11,39)(12,42)(13,23)(15,48)(16,45)(17,46)(19,52)(22,43)(24,44)(25,54)(27,55)(28,47)(29,35)(31,59)(32,61)(33,62)(34,63)(36,57)(37,56)(38,50)(40,51)(41,58)(49,53)(60,64);(2,10)(4,20,21,23)(6,24)(8,31)(9,35,18,42)(12,29)(14,45,46,47)(15,49)(19,53,54,48)(22,44,55,26)(25,52)(27,43)(30,58,59,62)(33,41)(34,61,50,64)(36,39,51,56)

Polycyclic presentation:

2^9|b^2=fh,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,e^c=ei,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320520, #72320552, #72320706
maximal normal subgroup of	#72320520, #72320552, #72320706
maximal quotient of	#72320963, #72321008, #72321110, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406798);