#7233784

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#35927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1907 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,28)(19,29,33,24)(20,26,36,42)(21,27,23,38)(25,37,43,35)(30,40,31,44)(32,46,34,41)(39,47,45,48);(2,5)(3,7)(10,14)(11,12)(17,19)(18,24)(20,30)(21,32)(22,33)(23,34)(25,39)(26,44)(27,41)(28,29)(31,36)(35,47)(37,48)(38,46)(40,42)(43,45);(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,39)(26,38)(27,42)(28,43)(29,45)(31,33)(32,47)(34,48)(40,41)(44,46);(1,3,4,7)(2,6,5,8)(10,14)(15,16)(17,20)(18,26)(19,31)(21,37)(22,36)(23,35)(24,40)(25,38)(27,43)(28,42)(29,44)(30,33)(32,47)(34,48)(39,41)(45,46);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,28,38)(19,32,33,34)(20,35,36,37)(24,41,29,46)(25,42,43,26)(30,47,31,48)(39,40,45,44)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7140980);