#8687341

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

4081 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8594537);