#297158

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
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×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

759 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=h,c^2=fh,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=di,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204354);