#323397

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
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

608 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)(9,10,14,16,13,17,12,18)(11,15,21,22,20,23,19,24)(25,26,29,33,28,32,35,39)(27,34,37,40,36,30,38,31);(1,3,5,4)(2,6,8,7)(9,11,13,20)(10,15,17,23)(12,19,14,21)(16,22,18,24)(25,27,35,38,28,36,29,37)(26,30,39,40,32,34,33,31);(10,16)(12,14)(15,22)(17,18)(19,21)(23,24)(25,27,28,36)(26,31,32,40)(29,38,35,37)(30,39,34,33);(1,4,5,3)(2,7,8,6)(10,17)(15,23)(16,18)(22,24)

Polycyclic presentation:

2^9|a^2=efh,b^2=eghi,c^2=i,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,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, 230593);