#323298

order	512
nilpotency class	3
isoclinism	isoclinic class of #298603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

628 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)(9,10,14,16,12,17,13,18)(11,19,22,24,20,15,21,23)(25,26,30,35,28,33,29,34)(27,36,39,32,37,31,38,40);(1,3,5,4)(2,6,8,7)(9,11,12,20)(10,15,17,19)(13,21,14,22)(16,23,18,24)(25,27,29,38,28,37,30,39)(26,31,34,32,33,36,35,40);(10,16)(13,14)(15,23)(17,18)(19,24)(21,22)(25,27,28,37)(26,32,33,40)(29,39,30,38)(31,35,36,34);(1,4,5,3)(2,7,8,6)(10,17)(15,19)(16,18)(23,24)(26,33)(31,36)(32,40)(34,35)

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=i,d^2=eh,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, 230494);