#8258567

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

4171 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,24)(20,26)(21,28)(22,27)(23,25)(29,32)(30,31)(33,34,40,44)(35,45,47,41)(36,37,38,42)(39,43,48,46);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,19)(18,24)(20,22)(21,25)(23,28)(26,27)(29,30)(31,32)(33,35,40,47)(34,41,44,45)(36,46,38,43)(37,48,42,39);(4,12)(7,14)(10,16)(13,15)(18,25)(19,28)(22,29)(26,31)(33,36,40,38)(34,42,44,37)(35,46,47,43)(39,45,48,41);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20)(18,26)(19,29)(21,27)(22,28)(23,30)(24,32)(25,31)(33,37)(34,36)(35,39)(38,44)(40,42)(41,43)(45,46)(47,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,21)(18,19)(20,27)(22,31)(23,24)(25,28)(26,29)(30,32)(33,38,40,36)(34,37,44,42)(35,46,47,43)(39,45,48,41);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,22,23,29)(18,27,25,32)(19,20,28,30)(21,31,24,26)(33,39)(34,43)(35,37)(36,41)(38,45)(40,48)(42,47)(44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8165763);