#500732

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
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

1023 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)(5,6,13,20)(7,8,24,27)(9,18,29,35)(10,19,12,17)(11,16,28,21)(14,32,26,23)(15,34,30,25)(22,31,36,33);(5,7,10,23,13,24,12,32)(6,14,17,8,20,26,19,27)(9,22,11,30,29,36,28,15)(16,33,18,25,21,31,35,34);(2,4)(5,8)(6,15)(7,21)(9,25)(10,26)(11,31)(12,14)(13,27)(16,24)(17,22)(18,32)(19,36)(20,30)(23,35)(28,33)(29,34);(5,9,12,28,13,29,10,11)(6,16,19,35,20,21,17,18)(7,22,32,15,24,36,23,30)(8,25,14,33,27,34,26,31)

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=ei,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407928);