#7201054

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#52956	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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)(26,30)(27,33)(29,36)(31,39)(34,40)(35,37);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,26)(27,34)(28,30)(29,32)(31,35)(33,39)(36,38)(37,40);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27,28,35)(26,31,30,34)(29,37,36,33)(32,39,38,40);(2,6)(3,7)(11,19)(14,17)(18,24)(21,23)(25,26)(27,31)(28,30)(29,32)(33,40)(34,35)(36,38)(37,39);(3,7)(5,8)(10,16)(14,17)(15,22)(21,23)(26,30)(31,34)(32,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7108250);