#499771

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

Polycyclic presentation:

2^9|c^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, 406967);