#7092889

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1381 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,30,25)(20,26,34,41)(21,27,22,24)(31,43,44,40)(32,39,35,37)(33,42,36,38)(45,48,46,47);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,30)(20,33)(21,22)(25,39)(26,41)(27,28)(29,37)(31,45)(34,36)(44,46)(47,48);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,31)(21,35)(22,32)(23,30)(24,37)(26,40)(27,39)(28,29)(33,46)(34,44)(36,45)(38,47)(41,43)(42,48);(11,16)(13,15)(17,20,23,34)(18,26,28,41)(19,31,30,44)(21,33,22,36)(24,38,27,42)(25,40,29,43)(32,45,35,46)(37,47,39,48);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,21,23,22)(18,24,28,27)(19,32,30,35)(20,33,34,36)(25,39,29,37)(26,38,41,42)(31,45,44,46)(40,48,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

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, 7000085);