#299493

order	512
nilpotency class	3
isoclinism	isoclinic class of #299469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,g^b=gh,d^c=di,g^c=gh

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, 206689);