#111568

order	512
nilpotency class	3
isoclinism	isoclinic class of #9091
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,g^b=gh,f^c=fh

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, 18764);