#7302621

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	#45091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1656 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7209817);