#10069039

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

4116 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9976235);