#7206229

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
prime radical	#30513	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1576 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,27)(21,36)(22,28)(23,38)(24,29)(26,35)(30,40)(31,45)(32,41)(33,47)(34,42)(37,43)(39,44)(46,48);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,19,24,34)(18,25,29,42)(20,30,21,31)(22,32,23,33)(26,39,43,48)(27,40,36,45)(28,41,38,47)(35,44,37,46);(11,15)(13,16)(17,20,24,21)(18,26,29,43)(19,30,34,31)(22,35,23,37)(25,39,42,48)(27,28,36,38)(32,44,33,46)(40,41,45,47);(1,4,6,5)(2,7,3,8)(10,14)(11,15)(17,21,24,20)(18,27,29,36)(19,31,34,30)(22,35,23,37)(25,40,42,45)(26,28,43,38)(32,44,33,46)(39,41,48,47);(1,5,6,4)(2,7,3,8)(11,15)(13,16)(17,22,24,23)(18,28,29,38)(19,32,34,33)(20,35,21,37)(25,41,42,47)(26,36,43,27)(30,44,31,46)(39,45,48,40)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=fi,e^2=fi,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, 7113425);