#9913289

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

4227 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)(3,11)(4,6)(5,8)(7,9)(10,16)(12,15)(13,14)(17,18,23,21)(19,24,20,25)(22,26,30,29)(27,31,28,32)(33,34,38,42)(35,39,44,48)(36,45,46,40)(37,41,43,47);(1,3,7,5)(2,8,9,11)(4,12,10,14)(6,13,16,15)(18,21)(24,25)(26,29)(31,32)(33,35)(34,39)(36,37)(38,44)(40,47)(41,45)(42,48)(43,46);(2,9)(4,10)(8,11)(12,14)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,36,38,46)(34,40,42,45)(35,43,44,37)(39,41,48,47);(1,4,7,10)(2,6,9,16)(3,12,5,14)(8,13,11,15)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,36)(34,40)(35,37)(38,46)(39,47)(41,48)(42,45)(43,44);(1,5,7,3)(2,8,9,11)(4,14,10,12)(6,13,16,15)(17,21,23,18)(19,25,20,24)(22,26,30,29)(27,31,28,32)(33,37)(34,41)(35,36)(38,43)(39,45)(40,48)(42,47)(44,46);(1,6)(2,10)(3,13)(4,9)(5,15)(7,16)(8,14)(11,12)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,43)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820485);