#8603530

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

4192 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8510726);