#9835882

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

4264 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)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,18)(19,26)(20,28)(21,31)(22,24)(23,29)(25,30)(27,32)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,32)(30,31)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,6)(7,14)(8,11)(15,16)(18,24)(20,30)(23,26)(27,31)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,20,22,30)(18,25,24,28)(19,27,29,31)(21,23,32,26)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,19)(18,26)(20,27)(21,25)(22,29)(23,24)(28,32)(30,31)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,21)(18,27)(19,28)(20,23)(22,32)(24,31)(25,29)(26,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,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, 9743078);