#9719166

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

3943 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)(3,8)(4,13)(5,10)(6,14)(7,9)(11,16)(12,15)(17,18,23,25)(19,24,22,32)(20,26,29,21)(27,28,30,31)(33,34)(35,42)(36,40)(37,41)(38,43)(39,46)(44,48)(45,47);(1,3,7,11)(2,8,9,16)(4,5,14,15)(6,12,13,10)(17,19,23,22)(18,24,25,32)(20,27,29,30)(21,31,26,28)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,48)(42,43);(2,9)(3,11)(4,14)(10,12)(18,25)(20,29)(24,32)(27,30)(33,36,38,37)(34,40,43,41)(35,44,46,45)(39,47,42,48);(1,4)(2,6)(3,5)(7,14)(8,12)(9,13)(10,16)(11,15)(17,20)(18,26)(19,27)(21,25)(22,30)(23,29)(24,28)(31,32)(33,37,38,36)(34,41,43,40)(35,45,46,44)(39,48,42,47);(1,5)(2,10)(3,4)(6,16)(7,15)(8,13)(9,12)(11,14)(17,21,23,26)(18,20,25,29)(19,28,22,31)(24,30,32,27)(33,35)(34,42)(36,44)(37,45)(38,46)(39,43)(40,48)(41,47);(1,6)(2,4)(3,12)(5,8)(7,13)(9,14)(10,11)(15,16)(17,22,23,19)(18,24,25,32)(20,30,29,27)(21,31,26,28)(33,34)(35,39)(36,40)(37,41)(38,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9626362);