#10146876

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

4192 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,23)(19,21,25,31)(20,24,29,32)(26,30,27,28)(33,34,40,38)(35,43,39,41)(36,42,37,46)(44,48,45,47);(3,9)(7,13)(10,15)(14,16)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,35,40,39)(34,41,38,43)(36,44,37,45)(42,47,46,48);(2,6)(7,13)(8,11)(14,16)(18,23)(19,25)(24,32)(26,27)(33,36,40,37)(34,42,38,46)(35,44,39,45)(41,47,43,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,37,40,36)(34,42,38,46)(35,45,39,44)(41,47,43,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20,22,29)(18,24,23,32)(19,26,25,27)(21,30,31,28)(33,38,40,34)(35,43,39,41)(36,46,37,42)(44,48,45,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,19)(20,28)(22,31)(23,25)(24,27)(26,32)(29,30)(33,39,40,35)(34,41,38,43)(36,45,37,44)(42,47,46,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=g,e^2=gi,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, 10054072);