#7326191

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=i,e^2=i,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^c=ei,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, 7233387);