#6564720

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#3322	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1906 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6471916);