#8630908

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

4193 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,e^2=ghi,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, 8538104);