#8757286

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

4117 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,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, 8664482);