#9913234

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

4217 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)(19,21)(20,25)(22,26)(23,24)(27,29)(28,30)(31,32)(33,34)(35,40)(36,44)(37,38)(39,43)(41,47)(42,48)(45,46);(1,3,7,5)(2,8,9,11)(4,12,10,14)(6,13,16,15)(18,24)(21,25)(26,32)(29,30)(33,35)(34,40)(36,37)(38,44)(39,45)(41,48)(42,47)(43,46);(2,9)(4,10)(8,11)(12,14)(17,19,23,20)(18,21,24,25)(22,27,31,28)(26,29,32,30)(33,36,39,47)(34,41,43,44)(35,42,45,37)(38,46,48,40);(1,4,7,10)(2,6,9,16)(3,12,5,14)(8,13,11,15)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,30,32,29)(33,36)(34,41)(35,37)(38,46)(39,47)(40,48)(42,45)(43,44);(1,5,7,3)(2,8,9,11)(4,14,10,12)(6,13,16,15)(17,21,23,25)(18,19,24,20)(22,30,31,29)(26,28,32,27)(33,37)(34,38)(35,36)(39,42)(40,44)(41,46)(43,48)(45,47);(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)(23,31)(24,32)(25,30)(33,38)(34,42)(35,44)(36,46)(37,43)(39,48)(40,47)(41,45)

Polycyclic presentation:

2^9|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, 9820430);