#292700

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

898 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)(3,7)(4,6)(5,8)(10,13)(11,15)(14,16)(17,18,22,26)(19,27,29,24)(20,23,25,21)(28,32,31,30);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,16,15)(17,19,22,29)(18,24,26,27)(20,30,25,32)(21,28,23,31);(9,11,13,15,12,16,10,14)(17,20,23,26,22,25,21,18)(19,27,28,30,29,24,31,32);(2,6)(5,7)(9,10,12,13)(11,15,16,14)(17,21,22,23)(18,25,26,20)(19,28,29,31)(24,32,27,30)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=dhi

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, 199896);