#10040996

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

4087 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9948192);