#501240

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

Polycyclic presentation:

2^9|a^2=f,b^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, 408436);