#8496894

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

4160 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8404090);