#318901

order	512
nilpotency class	3
isoclinism	isoclinic class of #262787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

566 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egh,c^2=i,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=di,f^c=fh,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, 226097);