#329331

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

688 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236527);