#10042540

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

4050 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=gi,e^2=g,f^2=h,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, 9949736);