#7337156

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324820
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	#53073	C₂
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)(4,7)(5,8)(17,18,22,26)(19,27,28,40)(20,24,31,38)(21,33,34,25)(23,35,36,30)(29,41,42,48)(32,45,44,39)(37,46,47,43);(1,3,6,2)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(19,28)(20,31)(23,36)(24,38)(27,40)(30,35)(32,44)(39,45);(9,11)(10,13)(12,16)(14,15)(17,19,21,30,22,28,34,35)(18,23,25,27,26,36,33,40)(20,29,32,43,31,42,44,46)(24,37,39,41,38,47,45,48);(1,4,6,5)(2,7,3,8)(10,14)(11,16)(17,20)(18,24)(19,29)(21,32)(22,31)(23,37)(25,39)(26,38)(27,41)(28,42)(30,43)(33,45)(34,44)(35,46)(36,47)(40,48);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(20,31)(24,38)(29,42)(32,44)(37,47)(39,45)(41,48)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7244352);