#352539

order	512
nilpotency class	3
isoclinism	isoclinic class of #347758
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

615 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=f,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,c^b=ci,g^d=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259735);