#10040428

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

4001 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)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,22,24)(19,23,26,21)(20,28,30,25)(27,31,29,32)(33,34,38,36)(35,40,43,39)(37,44,45,41)(42,46,48,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,22,26)(18,23,24,21)(20,29,30,27)(25,31,28,32)(33,35,38,43)(34,39,36,40)(37,42,45,48)(41,46,44,47);(2,7)(6,13)(8,11)(14,15)(18,24)(19,26)(20,30)(31,32)(33,35,38,43)(34,39,36,40)(37,42,45,48)(41,46,44,47);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,22,30)(18,25,24,28)(19,27,26,29)(21,31,23,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,19)(18,23)(20,27)(21,24)(22,26)(25,32)(28,31)(29,30)(33,36,38,34)(35,39,43,40)(37,41,45,44)(42,47,48,46);(1,4,5,12)(2,8,7,11)(3,9,10,16)(6,14,13,15)(17,21,22,23)(18,26,24,19)(20,31,30,32)(25,29,28,27)(33,37,38,45)(34,41,36,44)(35,42,43,48)(39,46,40,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=i,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, 9947624);