#6644457

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1279 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=efhi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6551653);