#352676

order	512
nilpotency class	3
isoclinism	isoclinic class of #352019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=fi,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=dh

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, 259872);