#7184843

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1393 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 7092039);