#8633578

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

4093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=g,f^2=hi,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, 8540774);