#7078648

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

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=eh,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, 6985844);