#304931

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

668 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)(9,10,13,16)(11,19,20,15)(12,22,21,24)(14,18,23,17)(25,26)(27,31)(28,36)(29,32)(30,34)(33,38)(35,39)(37,40);(1,3,5,4)(2,6,8,7)(9,11,13,20)(10,15,16,19)(12,23,21,14)(17,22,18,24)(25,27)(26,31)(28,33)(29,35)(30,37)(32,39)(34,40)(36,38);(1,3,5,4)(2,7,8,6)(9,12,23,11,13,21,14,20)(10,15,18,24,16,19,17,22)(25,28,38,40,30,39,29,31)(26,27,32,35,34,37,33,36);(1,4,5,3)(2,7,8,6)(10,16)(15,19)(17,18)(22,24)(25,29,30,38)(26,32,34,33)(27,35,37,36)(28,31,39,40)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=efi,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212127);