#9814620

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

4093 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)(3,5)(4,9)(6,15)(7,8)(10,12)(11,14)(13,16)(17,18,23,21)(19,27,29,24)(20,30,31,25)(22,32,28,26)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(1,3)(2,5)(4,12)(6,13)(7,11)(8,14)(9,10)(15,16)(17,19,23,29)(18,24,21,27)(20,28,31,22)(25,26,30,32)(35,41)(39,42)(43,45)(47,48);(2,8)(3,11)(4,13)(10,15)(17,20,23,31)(18,25,21,30)(19,22,29,28)(24,32,27,26)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,4,7,13)(2,9,8,16)(3,12,11,6)(5,10,14,15)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,26)(27,32)(33,36,38,34)(35,42,41,39)(37,44,46,40)(43,48,45,47);(1,5)(2,3)(4,10)(6,16)(7,14)(8,11)(9,12)(13,15)(17,21,23,18)(19,24,29,27)(20,30,31,25)(22,32,28,26)(33,35)(34,39)(36,42)(37,45)(38,41)(40,48)(43,46)(44,47);(1,6,7,12)(2,10,8,15)(3,13,11,4)(5,9,14,16)(17,22,23,28)(18,26,21,32)(19,20,29,31)(24,30,27,25)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,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, 9721816);