#9494736

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

4001 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,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, 9401932);