#10055867

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

4232 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=gh,e^2=gi,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, 9963063);