#7088201

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1234 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,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=ehi,f^b=fi,d^c=dh,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, 6995397);