#10047205

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

3964 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=gi,e^2=g,f^2=gh,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, 9954401);