#8533303

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)(3,4)(5,10)(6,7)(8,9)(11,12)(13,16)(14,15)(17,18)(19,24)(20,26)(21,22)(23,25)(27,29)(28,30)(31,32)(33,34,40,44)(35,45,48,41)(36,42,37,38)(39,43,47,46);(2,7)(3,9)(10,15)(11,13)(17,19,23,28)(18,24,25,30)(20,29,31,21)(22,26,27,32)(33,35)(34,41)(36,43)(37,46)(38,47)(39,42)(40,48)(44,45);(2,7)(4,8)(10,15)(12,16)(18,25)(19,28)(21,29)(26,32)(33,36,40,37)(34,42,44,38)(35,43,48,46)(39,45,47,41);(1,3,6,9)(2,8,7,4)(5,13,14,11)(10,12,15,16)(17,20)(18,26)(19,21)(22,24)(23,31)(25,32)(27,30)(28,29)(33,37,40,36)(34,42,44,38)(35,46,48,43)(39,45,47,41);(1,4,6,8)(2,9,7,3)(5,12,14,16)(10,13,15,11)(17,21)(18,27)(19,20)(22,25)(23,29)(24,32)(26,30)(28,31)(33,38,40,42)(34,36,44,37)(35,47,48,39)(41,43,45,46);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,16)(9,13)(17,22,23,27)(18,21,25,29)(19,26,28,32)(20,30,31,24)(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=g,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, 8440499);