#323291

order	512
nilpotency class	3
isoclinism	isoclinic class of #298603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,8)(3,7,4,6)(9,10,14,22,13,17,12,16)(11,18,21,15,19,24,20,23)(25,26,29,32)(27,30,36,40)(28,31,33,37)(34,38,35,39);(1,3,5,4)(2,6,8,7)(11,19)(15,23)(18,24)(20,21)(25,27,33,34,29,36,28,35)(26,30,37,38,32,40,31,39);(9,11,13,19)(10,15,17,23)(12,21,14,20)(16,24,22,18)(27,34)(28,33)(30,38)(31,37)(35,36)(39,40);(1,4,5,3)(2,7,8,6)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230487);