#501237

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1049 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)(5,6,10,15)(7,8,21,24)(9,19,29,36)(11,16,28,17)(12,32,33,20)(13,18,27,23)(14,34,35,30)(22,25,31,26);(5,7)(6,12)(8,19)(9,27)(10,21)(11,31)(13,29)(14,26)(15,33)(16,18)(17,23)(20,30)(22,28)(24,36)(25,35)(32,34);(2,4)(5,6)(7,18)(8,13)(9,14)(10,15)(11,17)(12,31)(16,28)(19,30)(20,26)(21,23)(22,33)(24,27)(25,32)(29,35)(34,36);(5,8,10,24)(6,13,15,27)(7,19,21,36)(9,12,29,33)(11,25,28,26)(14,31,35,22)(16,20,17,32)(18,30,23,34)

Polycyclic presentation:

2^9|a^2=f,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, 408433);