#10422477

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

4269 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)(3,9)(4,8)(5,7)(6,13)(10,14)(11,15)(12,16)(17,18,24,27)(19,28,22,25)(20,23,21,26)(29,30,31,32)(34,39)(36,43)(41,45)(44,48);(2,5)(8,9)(10,13)(15,16)(17,19,24,22)(18,25,27,28)(20,31,21,29)(23,30,26,32)(33,34)(35,36)(37,41)(38,39)(40,43)(42,44)(45,46)(47,48);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,20,24,21)(18,26,27,23)(19,29,22,31)(25,30,28,32)(33,35,38,40)(34,36,39,43)(37,42,46,47)(41,44,45,48);(1,4,7,3)(2,9,5,8)(6,11,14,12)(10,15,13,16)(17,21,24,20)(18,23,27,26)(19,29,22,31)(25,30,28,32)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,48)(44,47);(1,5,7,2)(3,9,4,8)(6,13,14,10)(11,16,12,15)(17,22,24,19)(18,25,27,28)(20,31,21,29)(23,32,26,30)(33,36)(34,40)(35,39)(37,44)(38,43)(41,47)(42,45)(46,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,23)(18,20)(19,30)(21,27)(22,32)(24,26)(25,31)(28,29)(33,37,38,46)(34,41,39,45)(35,42,40,47)(36,44,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10329673);