#499774

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

1253 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)(8,24)(9,20)(10,16)(11,32)(12,17)(13,21)(14,25)(15,30)(19,27)(22,33)(23,26)(28,34)(29,35)(31,36);(5,7,12,24)(6,13,17,28)(8,20,18,36)(9,27,31,14)(10,23,11,22)(15,33,35,26)(16,19,32,25)(21,29,34,30);(2,4)(5,6)(7,19)(8,14)(9,15)(10,32)(11,16)(12,17)(13,23)(18,27)(20,30)(21,26)(22,28)(24,25)(29,36)(31,35)(33,34);(5,8,12,18)(6,14,17,27)(7,20,24,36)(9,28,31,13)(10,26,11,33)(15,22,35,23)(16,34,32,21)(19,30,25,29)

Polycyclic presentation:

2^9|b^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, 406970);