#9506833

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

4194 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18)(19,22)(20,24)(21,23)(25,26)(27,30)(28,29)(31,32)(33,34,38,36)(35,41,43,39)(37,44,46,40)(42,47,45,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,32)(29,31)(34,36)(39,41)(40,44)(47,48);(2,7)(6,13)(8,11)(14,15)(18,23)(19,26)(20,28)(30,31)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,21,28)(18,24,23,29)(19,27,26,32)(22,30,25,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,18)(19,25)(20,29)(21,23)(22,26)(24,28)(27,30)(31,32)(33,36,38,34)(35,39,43,41)(37,44,46,40)(42,47,45,48);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,19)(18,25)(20,27)(21,26)(22,23)(24,31)(28,32)(29,30)(33,37,38,46)(34,40,36,44)(35,42,43,45)(39,47,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=i,e^2=g,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, 9414029);