#7012283

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1386 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,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, 6919479);