#9814721

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

3974 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)(4,5)(6,7)(8,12)(10,15)(11,16)(13,14)(17,18)(19,24)(20,26)(21,27)(22,29)(23,25)(28,32)(30,31)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(2,7)(4,13)(8,9)(10,16)(17,19,23,20)(18,24,25,26)(21,29,31,28)(22,30,32,27)(35,41)(39,42)(43,45)(47,48);(18,25)(19,20)(27,30)(28,29)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,3,6,12)(2,8,7,9)(4,10,13,16)(5,11,14,15)(17,20,23,19)(18,26,25,24)(21,28,31,29)(22,27,32,30)(33,36,38,34)(35,42,41,39)(37,44,46,40)(43,48,45,47);(1,4,6,13)(2,5,7,14)(3,10,12,16)(8,11,9,15)(17,21)(18,27)(19,29)(20,28)(22,24)(23,31)(25,30)(26,32)(33,35)(34,39)(36,42)(37,45)(38,41)(40,48)(43,46)(44,47);(1,5)(2,4)(3,11)(6,14)(7,13)(8,10)(9,16)(12,15)(17,22,23,32)(18,28,25,29)(19,30,20,27)(21,26,31,24)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gi,e^2=g,f^2=h,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, 9721917);