#499776

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=hi

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, 406972);