#10247604

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

4054 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154800);