#8434709

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

4094 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)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(20,28)(24,31)(27,32)(29,30)(34,37)(36,40)(41,45)(44,46);(3,10)(6,7)(11,16)(13,14)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,34,39,37)(35,40,43,36)(38,41,47,45)(42,46,48,44);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,35,39,43)(34,40,37,36)(38,42,47,48)(41,46,45,44);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36)(34,35)(37,43)(38,46)(39,40)(41,48)(42,45)(44,47);(1,4,5,12)(2,8,9,15)(3,11,10,16)(6,13,7,14)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,37,39,34)(35,36,43,40)(38,45,47,41)(42,44,48,46);(1,2)(3,6)(4,8)(5,9)(7,10)(11,13)(12,15)(14,16)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,38,39,47)(34,41,37,45)(35,42,43,48)(36,44,40,46)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8341905);