#507793

order	512
nilpotency class	4
isoclinism	isoclinic class of #499904
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3192
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#29409
prime radical	#13309	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

507 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,9)(4,19)(5,11)(6,28)(7,13)(8,30)(10,21)(12,25)(14,42)(15,33)(16,47)(17,36)(18,50)(20,53)(22,52)(23,55)(24,34)(26,40)(27,56)(29,41)(31,37)(32,44)(35,46)(38,54)(39,57)(43,60)(45,59)(48,64)(49,62)(51,63)(58,61);(1,3)(2,9)(4,20)(5,24)(6,16)(7,17)(8,18)(10,32)(11,34)(12,35)(13,36)(14,22)(15,26)(19,42)(21,43)(23,51)(25,46)(27,49)(28,47)(29,58)(30,50)(31,45)(33,40)(37,63)(38,60)(39,61)(41,64)(44,54)(48,57)(52,53)(55,59)(56,62);(1,4)(2,10)(3,14)(5,21)(6,29)(7,22)(8,31)(9,32)(11,37)(12,41)(13,38)(15,43)(16,48)(17,20)(18,51)(19,28)(23,25)(24,44)(26,54)(27,57)(30,39)(33,59)(34,63)(35,64)(36,60)(40,55)(42,47)(45,46)(49,58)(50,61)(52,56)(53,62);(1,5,25,6,7,26,8,27)(2,11,12,28,13,40,30,56)(3,15,46,49,17,24,18,16)(4,21,23,29,22,54,31,57)(9,33,35,62,36,34,50,47)(10,37,41,19,38,55,39,52)(14,43,45,58,20,44,51,48)(32,59,64,53,60,63,61,42)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414989);