#9847655

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

4277 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9754851);