#323568

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,8)(3,7,4,6)(9,10,13,14,12,16,11,15)(17,18,21,25,20,24,27,23)(19,26,30,32,29,22,28,31);(1,3,5,4)(2,6,8,7)(17,19,27,28,20,29,21,30)(18,22,23,32,24,26,25,31);(9,10,12,16)(11,14,13,15)(18,23)(19,28)(21,27)(22,26)(24,25)(29,30);(1,4,5,3)(2,7,8,6)(9,11,12,13)(10,14,16,15)(19,29)(22,26)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=efh,b^2=eg,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230764);