#8635457

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

4026 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=g,e^2=gh,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, 8542653);