#10534392

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

4134 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,16)(14,15)(18,22)(20,28)(23,25)(26,29)(33,34,40,42)(35,43,38,39)(36,45,37,41)(44,47,46,48);(2,7)(3,9)(10,15)(11,16)(18,22)(23,25)(24,30)(31,32)(33,35,40,38)(34,39,42,43)(36,44,37,46)(41,47,45,48);(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,36,40,37)(34,41,42,45)(35,44,38,46)(39,47,43,48);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,13,15,12)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,37,40,36)(34,41,42,45)(35,44,38,46)(39,48,43,47);(1,4,6,8)(2,9,7,3)(5,13,14,12)(10,11,15,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,28)(30,31)(33,38,40,35)(34,39,42,43)(36,46,37,44)(41,47,45,48);(1,5,6,14)(2,10,7,15)(3,11,9,16)(4,12,8,13)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29)(33,39)(34,38)(35,42)(36,47)(37,48)(40,43)(41,46)(44,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10441588);