#512208

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

616 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=e,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,h^a=hi,c^b=ci,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419404);