#9508803

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

4005 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34)(35,43)(36,45)(37,42)(38,46)(39,41)(40,44)(47,48);(2,6)(7,9)(8,12)(14,15)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(33,35,39,44)(34,40,41,43)(36,37,46,47)(38,42,45,48);(2,6)(3,10)(4,13)(14,15)(18,19)(20,27)(21,29)(30,31)(34,41)(35,44)(36,46)(42,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,36)(34,38)(35,37)(39,46)(40,42)(41,45)(43,48)(44,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,37)(34,42)(35,36)(38,40)(39,47)(41,48)(43,45)(44,46);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,38)(34,36)(35,42)(37,40)(39,45)(41,46)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=gi,f^2=gh,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, 9415999);