#9267034

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

3994 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=gi,e^2=gh,f^2=g,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, 9174230);