#8634803

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

4184 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=h,e^2=gh,f^2=gi,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, 8541999);