#507548

order	512
nilpotency class	4
isoclinism	isoclinic class of #499903
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	#29237
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

498 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 136 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)(5,23)(6,12)(7,27)(9,21)(10,26)(11,13)(14,40)(15,43)(16,32)(17,47)(19,52)(20,50)(22,35)(24,34)(25,49)(28,51)(29,41)(30,46)(31,38)(33,54)(36,39)(37,55)(42,59)(44,58)(45,63)(48,64)(53,57)(56,61)(60,62);(1,3,13,38,6,16,7,17)(2,8,11,31,12,32,27,47)(4,19,22,48,20,14,28,42)(5,15,39,62,25,45,26,46)(9,29,37,61,33,57,34,58)(10,30,23,43,36,60,49,63)(18,40,35,59,50,52,51,64)(21,53,55,44,54,41,24,56);(1,4,6,20)(2,9)(3,14,16,19)(5,24,25,55)(7,28,13,22)(8,29)(10,35)(11,37)(12,33)(15,44,45,56)(17,48,38,42)(18,49)(21,26,54,39)(23,50)(27,34)(30,59)(31,61)(32,57)(36,51)(40,63)(41,46,53,62)(43,52)(47,58)(60,64);(5,25)(10,36)(15,45)(18,50)(21,54)(23,49)(24,55)(26,39)(30,60)(35,51)(40,52)(41,53)(43,63)(44,56)(46,62)(59,64)

Polycyclic presentation:

2^9|b^2=egh,c^2=g,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,e^c=ei,f^c=fhi,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, 414744);