#7326010

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
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

1482 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)(17,18,22,26)(19,27,31,43)(20,24,33,40)(21,35,28,25)(23,36,38,30)(29,42,32,46)(34,48,44,41)(37,47,39,45);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16);(9,11)(10,13)(12,16)(14,15)(17,19,28,36,22,31,21,30)(18,23,35,43,26,38,25,27)(20,32,44,45,33,29,34,47)(24,39,48,42,40,37,41,46);(3,8)(5,7)(10,14)(11,16)(17,20,22,33)(18,24,26,40)(19,29,31,32)(21,34,28,44)(23,37,38,39)(25,41,35,48)(27,42,43,46)(30,45,36,47);(2,6)(5,7)(10,14)(13,15)(20,33)(24,40)(29,32)(34,44)(37,39)(41,48)(42,46)(45,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=gi,d^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=eh,d^c=dhi,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, 7233206);