#7125155

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1409 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7032351);