#9835834

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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=hi,e^2=gh,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, 9743030);