#8461980

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

4193 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18)(19,29)(20,26)(21,23)(22,32)(24,28)(25,31)(27,30)(36,44)(41,45)(42,48)(46,47);(2,3)(7,9)(8,10)(14,15)(17,19,24,31)(18,25,28,29)(20,30,21,22)(23,27,26,32)(33,34,38,37)(35,40,43,39)(36,41,44,45)(42,47,48,46);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,20,24,21)(18,26,28,23)(19,30,31,22)(25,32,29,27)(33,35)(34,39)(36,42)(37,40)(38,43)(41,46)(44,48)(45,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,21,24,20)(18,26,28,23)(19,22,31,30)(25,32,29,27)(33,35)(34,40)(36,42)(37,39)(38,43)(41,47)(44,48)(45,46);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,22)(18,27)(19,20)(21,31)(23,29)(24,30)(25,26)(28,32)(33,36,38,44)(34,41,37,45)(35,42,43,48)(39,46,40,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,23,24,26)(18,21,28,20)(19,27,31,32)(22,29,30,25)(33,37,38,34)(35,40,43,39)(36,45,44,41)(42,47,48,46)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=g,e^2=hi,f^2=ghi,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, 8369176);