#6634866

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,34,30,33,27,31)(28,32,36,39,37,40,35,38);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,30,29)(26,31,33,34)(28,36,37,35)(32,39,40,38);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,31)(27,29)(32,38)(33,34)(35,36)(39,40);(3,7)(5,8)(10,16)(11,19)(15,22)(18,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,40)(34,39);(2,6)(5,8)(10,16)(14,17)(15,22)(21,23)(26,33)(28,37)(31,34)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6542062);