#10412258

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

4029 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10319454);