#8695870

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603066);