#8258579

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

4147 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=g,e^2=ghi,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, 8165775);