#330744

order	512
nilpotency class	3
isoclinism	isoclinic class of #329395
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×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

608 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237940);