#10181201

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

4077 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,27)(20,25)(21,28)(22,23)(24,29)(26,31)(30,32)(33,34)(35,44)(36,38)(37,41)(39,43)(40,45)(42,47)(46,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(18,23)(20,30)(24,27)(26,28)(35,45)(37,46)(40,44)(41,48);(2,3)(7,9)(8,10)(14,15)(17,19,22,29)(18,24,23,27)(20,28,30,26)(21,25,31,32)(33,35,39,45)(34,40,43,44)(36,46,42,37)(38,41,47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,24)(20,26)(21,25)(22,29)(23,27)(28,30)(31,32)(33,36,39,42)(34,38,43,47)(35,37,45,46)(40,48,44,41);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,20,22,30)(18,25,23,32)(19,28,29,26)(21,24,31,27)(33,37)(34,41)(35,36)(38,44)(39,46)(40,47)(42,45)(43,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,31)(18,26,23,28)(19,25,29,32)(20,27,30,24)(33,38)(34,42)(35,41)(36,43)(37,40)(39,47)(44,46)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10088397);