#7145459

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#42887	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1706 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)(4,5)(7,8)(11,15)(14,16)(17,18,23,29)(19,24,33,41)(20,26,35,44)(21,28,22,27)(25,37,43,34)(30,46,31,39)(32,45,36,40)(38,47,42,48);(2,5)(3,7)(10,13)(11,15)(17,19,23,33)(18,24,29,41)(20,31,35,30)(21,36,22,32)(25,42,43,38)(26,39,44,46)(27,45,28,40)(34,47,37,48);(9,10)(11,14)(12,13)(15,16)(17,20,23,35)(18,25,29,43)(19,30,33,31)(21,34,22,37)(24,38,41,42)(26,28,44,27)(32,47,36,48)(39,45,46,40);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,26)(19,31)(21,37)(22,34)(23,35)(24,39)(25,27)(28,43)(29,44)(30,33)(32,47)(36,48)(38,40)(41,46)(42,45);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,27,29,28)(19,32,33,36)(20,34,35,37)(24,40,41,45)(25,26,43,44)(30,47,31,48)(38,39,42,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^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=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7052655);