#8603560

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

4178 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)(3,9,11,7)(4,8,12,13)(10,15,16,14)(17,18,24,22)(19,23,27,25)(20,26,21,31)(28,30,29,32)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(2,6)(7,9)(8,13)(14,15)(17,19,24,27)(18,25,22,23)(20,28,21,29)(26,32,31,30)(35,42)(39,46)(43,47)(44,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,24,21)(18,26,22,31)(19,28,27,29)(23,30,25,32)(33,35,38,42)(34,39,41,46)(36,43,40,47)(37,44,45,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,21,24,20)(18,26,22,31)(19,29,27,28)(23,30,25,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,22,24,18)(19,25,27,23)(20,26,21,31)(28,30,29,32)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,23,24,25)(18,27,22,19)(20,30,21,32)(26,29,31,28)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8510756);