#8114381

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

3975 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)(5,12,13,8)(11,14,16,15)(18,20)(21,29)(23,26)(27,31)(33,34,38,39)(35,36,41,44)(37,40,45,46)(42,43,47,48);(2,4)(7,9)(8,12)(14,15)(18,20)(19,25)(24,28)(27,31)(34,39)(35,41)(40,46)(42,47);(3,10)(7,9)(11,16)(14,15)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,3,6,10)(2,7,4,9)(5,11,13,16)(8,14,12,15)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021577);