#6477701

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470355
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×#3188	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2007 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)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(22,27)(24,25)(28,35)(29,30)(31,36)(33,34);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,22,25,27,26,29,24,30)(23,31,34,36,32,28,33,35);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,23,26,32)(22,28,29,31)(24,33,25,34)(27,35,30,36);(7,14)(10,18)(13,15)(17,19)(22,29)(23,32)(27,30)(33,34);(6,12)(10,18)(11,20)(17,19)(21,24,26,25)(22,30,29,27)(23,33,32,34)(28,36,31,35)

Polycyclic presentation:

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

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, 6384897);