#8726696

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

4028 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=gi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8633892);