#323585

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
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

600 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)(9,10,12,16,13,15,21,23)(11,17,19,22,18,24,20,14)(25,26,30,32,28,33,29,34)(27,35,36,39,37,31,38,40);(1,3,5,4)(2,6,8,7)(11,18)(14,22)(17,24)(19,20)(25,27,29,38,28,37,30,36)(26,31,34,39,33,35,32,40);(9,11,13,18)(10,14,15,22)(12,20,21,19)(16,24,23,17)(26,32)(27,36)(29,30)(31,35)(33,34)(37,38);(1,4,5,3)(2,7,8,6)(10,15)(14,22)(16,23)(17,24)(27,37)(31,35)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=ef,b^2=egh,c^2=hi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,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, 230781);