#8728801

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

4151 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8635997);