#7326192

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	#45236	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 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)(17,18,21,23)(19,27,28,41)(20,25,32,36)(22,34,29,26)(24,37,35,31)(30,40,42,48)(33,45,43,39)(38,47,46,44);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(25,36)(26,34)(30,42)(31,37)(39,45)(44,47);(9,11)(10,13)(12,16)(14,15)(17,19,29,37,21,28,22,31)(18,24,34,41,23,35,26,27)(20,30,43,47,32,42,33,44)(25,38,45,48,36,46,39,40);(3,8)(5,7)(10,14)(11,16)(17,20)(18,25)(19,30)(21,32)(22,33)(23,36)(24,38)(26,39)(27,40)(28,42)(29,43)(31,44)(34,45)(35,46)(37,47)(41,48);(2,6)(5,7)(10,14)(13,15)(18,23)(19,28)(20,32)(26,34)(31,37)(33,43)(38,46)(40,48)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,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, 7233388);