#8393771

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

4134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300967);