#8825898

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

4285 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8733094);