#8463612

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

4046 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370808);