#9494715

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9401911);