#6766251

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,31,35)(27,32,33,39)(28,38,40,37)(29,34,30,36);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,32)(28,36)(29,38)(30,37)(31,33)(34,40)(35,39);(2,6)(3,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,30,32,31,40,29,39)(26,33,34,38,35,27,36,37);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,29,31,30)(26,34,35,36)(27,37,33,38)(28,39,40,32);(3,7)(5,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,31,30)(26,34,35,36)(27,38,33,37)(28,32,40,39)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgh,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=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, 6673447);