#7064093

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#42674	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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)(11,13)(12,14)(15,16)(17,18,22,27,21,26,35,24)(19,28,31,39,29,23,36,37)(20,32,34,40,33,25,41,42)(30,38,45,47,44,43,46,48);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,29)(18,23,26,28)(20,30,33,44)(22,31,35,36)(24,39,27,37)(25,38,32,43)(34,45,41,46)(40,48,42,47);(9,11,12,16)(10,13,14,15)(18,24)(19,29)(20,33)(22,35)(23,37)(25,42)(26,27)(28,39)(32,40)(38,48)(43,47)(45,46);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20)(18,25)(19,30)(21,33)(22,34)(23,38)(24,40)(26,32)(27,42)(28,43)(29,44)(31,45)(35,41)(36,46)(37,47)(39,48);(10,14)(13,15)(18,26)(19,29)(20,33)(24,27)(31,36)(34,41)(38,43)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6971289);