#10504984

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

4329 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=gh,e^2=gi,f^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, 10412180);