#7248067

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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	#42906	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1455 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:

(9,10)(11,15)(12,16)(13,14)(17,18,23,27,22,28,21,37)(19,29,32,42,33,24,38,43)(20,34,35,45,36,26,39,25)(30,46,48,40,31,41,47,44);(1,2)(3,7)(4,6)(5,8)(10,14)(11,12)(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);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,15)(13,14)(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)(5,7)(9,11,13,12)(10,15,14,16)(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);(2,6)(5,7)(9,12,13,11)(10,15,14,16)(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=g,b^2=i,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7155263);