#502013

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2218	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,9)(8,10)(11,12)(13,14)(15,23)(16,29)(17,20)(18,32)(19,27)(21,26)(22,30)(24,36)(25,38)(28,44)(31,33)(34,41)(35,40)(37,43)(39,42);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,24,40,18,28,16,27,17,25,26,43,30,42,31,41)(14,19,33,44,22,35,21,23,20,34,29,39,32,37,36,38);(2,4)(6,10)(9,12)(14,20)(15,25)(21,36)(22,32)(27,41)(28,42)(29,33)(40,43);(13,15,26,43,18,28,31,41,17,25,24,40,30,42,16,27)(14,19,29,39,22,35,36,38,20,34,33,44,32,37,21,23)

Polycyclic presentation:

2^9|b^2=efh,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409209);