#8993163

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

4153 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8900359);