#9910144

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

4251 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:

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

Polycyclic presentation:

2^9|a^2=h,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, 9817340);