#6541161

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×#3366	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=hi,e^2=h,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, 6448357);