#8533599

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

4189 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)(3,11)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(17,18)(19,20)(21,28)(22,25)(23,24)(26,27)(29,30)(31,32)(33,34,40,44)(35,45,48,41)(36,42,37,38)(39,43,47,46);(2,7)(3,12)(9,13)(10,16)(17,19,23,21)(18,20,24,28)(22,26,31,29)(25,27,32,30)(33,35)(34,41)(36,43)(37,46)(38,47)(39,42)(40,48)(44,45);(2,7)(4,14)(8,11)(10,16)(18,24)(19,21)(25,32)(26,29)(33,36,40,37)(34,42,44,38)(35,43,48,46)(39,45,47,41);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,13,15,9)(17,20)(18,19)(21,24)(22,30)(23,28)(25,29)(26,32)(27,31)(33,37,40,36)(34,42,44,38)(35,46,48,43)(39,45,47,41);(1,4)(2,9)(3,10)(5,8)(6,14)(7,13)(11,15)(12,16)(17,21,23,19)(18,20,24,28)(22,29,31,26)(25,27,32,30)(33,38,40,42)(34,36,44,37)(35,47,48,39)(41,43,45,46);(1,5)(2,10)(3,9)(4,8)(6,15)(7,16)(11,14)(12,13)(17,22,23,31)(18,25,24,32)(19,26,21,29)(20,27,28,30)(33,39,40,47)(34,43,44,46)(35,42,48,38)(36,45,37,41)

Polycyclic presentation:

2^9|a^2=ghi,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, 8440795);