#7078712

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 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)(17,18,23,27)(19,28,33,44)(20,25,35,43)(21,26,22,37)(24,31,40,32)(29,39,45,47)(30,38,46,48)(34,41,36,42);(9,10)(11,15)(12,14)(13,16)(17,19,23,33)(18,24,27,40)(20,29,35,45)(21,32,22,31)(25,38,43,48)(26,44,37,28)(30,34,46,36)(39,42,47,41);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,36)(22,34)(23,35)(24,38)(26,42)(27,43)(28,39)(30,32)(31,46)(33,45)(37,41)(40,48)(44,47);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20)(18,25)(19,30)(21,34)(22,36)(23,35)(24,39)(26,41)(27,43)(28,38)(29,32)(31,45)(33,46)(37,42)(40,47)(44,48);(10,14)(13,15)(17,21,23,22)(18,26,27,37)(19,31,33,32)(20,34,35,36)(24,28,40,44)(25,41,43,42)(29,30,45,46)(38,47,48,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dh,e^c=ei,f^d=fi

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, 6985908);