#7087161

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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 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)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,21,26)(20,25,34,41)(22,37,30,27)(24,36,39,32)(31,43,33,40)(35,47,44,42)(38,46,48,45);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,30,36,23,21,22,32)(18,24,37,26,28,39,27,29)(20,31,44,46,34,33,35,45)(25,38,47,40,41,48,42,43);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,31)(21,33)(22,35)(23,34)(24,38)(26,40)(27,42)(28,41)(29,43)(30,44)(32,45)(36,46)(37,47)(39,48);(11,16)(13,15)(17,21,23,19)(18,26,28,29)(20,33,34,31)(22,32,30,36)(24,27,39,37)(25,40,41,43)(35,45,44,46)(38,42,48,47);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(20,34)(25,41)(31,33)(35,44)(38,48)(40,43)(42,47)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,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=eh,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, 6994357);