#502516

order	512
nilpotency class	4
isoclinism	isoclinic class of #502469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

672 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:

(5,6,10,15,11,17,21,16)(7,18,23,13,24,32,30,33)(8,25,27,20,22,12,29,31)(9,28,26,14,19,34,36,35);(1,2)(3,4)(5,7,19,8,21,30,26,29,11,24,9,22,10,23,36,27)(6,12,14,18,15,31,34,13,17,25,35,32,16,20,28,33);(2,4)(5,6)(7,20)(8,13)(9,14)(10,16)(11,17)(12,30)(15,21)(18,27)(19,35)(22,33)(23,25)(24,31)(26,28)(29,32)(34,36);(5,8,26,24,10,27,19,30,11,22,36,7,21,29,9,23)(6,13,28,31,16,18,35,12,17,33,34,20,15,32,14,25)

Polycyclic presentation:

2^9|a^2=f,b^2=efh,d^2=eh,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409712);