#7207316

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,30)(20,36)(21,29)(22,28)(23,27)(24,34)(25,37)(26,38)(31,45)(32,41)(33,42)(35,43)(39,44)(40,48)(46,47);(1,3,6,2)(4,8,5,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,31)(21,33)(22,34)(23,35)(25,40)(26,45)(27,42)(28,30)(29,43)(32,38)(36,41)(37,47)(39,48)(44,46);(1,4,6,5)(2,7,3,8)(11,15)(13,16)(17,20)(18,25)(19,31)(21,37)(22,38)(23,39)(24,40)(26,29)(27,36)(28,44)(30,46)(32,34)(33,47)(35,48)(41,42)(43,45);(1,4,6,5)(2,8,3,7)(10,14)(11,15)(17,20)(18,26)(19,32)(21,39)(22,38)(23,37)(24,41)(25,29)(27,44)(28,36)(30,45)(31,34)(33,47)(35,48)(40,42)(43,46);(1,5,6,4)(2,8,3,7)(11,15)(13,16)(17,21,22,23)(18,27,28,29)(19,33,34,35)(20,37,38,39)(24,42,30,43)(25,36,44,26)(31,47,32,48)(40,41,46,45)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7114512);