#8664417

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

4249 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8571613);