#8695871

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

4199 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:

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

Polycyclic presentation:

2^9|a^2=ghi,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, 8603067);