#308988

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

588 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=f,c^2=f,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gi,d^c=dh,g^c=gi

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, 216184);