#7233748

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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

1915 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)(17,18,23,28)(19,24,31,43)(20,32,35,26)(21,27,22,38)(25,37,45,34)(29,46,33,40)(30,41,36,42)(39,47,44,48);(2,5)(3,7)(10,14)(11,12)(17,19)(18,24)(20,33)(21,36)(22,30)(23,31)(25,44)(26,46)(27,42)(28,43)(29,35)(32,40)(34,47)(37,48)(38,41)(39,45);(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,39)(26,27)(28,45)(30,47)(31,33)(32,38)(36,48)(40,42)(41,46)(43,44);(1,3,4,7)(2,6,5,8)(10,14)(15,16)(17,20)(18,26)(19,29)(21,37)(22,34)(23,35)(24,40)(25,27)(28,32)(30,48)(31,33)(36,47)(38,45)(39,42)(41,44)(43,46);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,27,28,38)(19,30,31,36)(20,34,35,37)(24,41,43,42)(25,26,45,32)(29,47,33,48)(39,46,44,40)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7140944);