#6564588

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37846	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6471784);