#10480947

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

4171 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)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31)(33,34,40,42)(35,43,38,39)(36,45,37,41)(44,47,46,48);(1,3,7,5)(2,8,10,4)(6,11,14,13)(9,15,16,12)(19,20)(23,24)(27,28)(31,32)(33,35,40,38)(34,39,42,43)(36,44,37,46)(41,47,45,48);(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,36,40,37)(34,41,42,45)(35,44,38,46)(39,47,43,48);(1,4,7,8)(2,3,10,5)(6,12,14,15)(9,11,16,13)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)(33,37,40,36)(34,41,42,45)(35,44,38,46)(39,48,43,47);(1,5,7,3)(2,8,10,4)(6,11,14,13)(9,12,16,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,38,40,35)(34,39,42,43)(36,46,37,44)(41,47,45,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,39)(34,38)(35,42)(36,47)(37,48)(40,43)(41,46)(44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10388143);