#9240986

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

4013 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)(3,9)(4,14)(5,10)(6,11)(7,8)(12,13)(15,16)(18,23)(19,26)(21,29)(28,31)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(2,8)(6,10)(9,14)(13,16)(17,18,22,23)(19,20,26,24)(21,25,29,30)(27,28,32,31)(35,43)(37,47)(39,48)(41,44);(1,3,7,4)(2,9,8,14)(5,15,11,12)(6,13,10,16)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,4,7,3)(2,9,8,14)(5,15,11,12)(6,16,10,13)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31);(1,5,7,11)(2,10,8,6)(3,12,4,15)(9,13,14,16)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,31)(30,32)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,6,7,10)(2,11,8,5)(3,13,4,16)(9,12,14,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9148182);