#8533598

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

4187 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

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, 8440794);