#10481053

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

4013 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)(3,10)(4,8)(5,7)(6,9)(11,15)(12,14)(13,16)(17,18,23,27)(19,25,26,22)(20,29,21,24)(28,30,31,32)(36,44)(39,45)(42,48)(46,47);(1,3,6,5)(2,7,9,10)(4,12,13,11)(8,15,16,14)(17,19)(18,22)(20,28)(21,31)(23,26)(24,30)(25,27)(29,32)(33,34)(35,40)(36,45)(37,41)(38,43)(39,44)(42,46)(47,48);(17,20,23,21)(18,24,27,29)(19,28,26,31)(22,30,25,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,48)(45,47);(1,4,6,13)(2,8,9,16)(3,11,5,12)(7,14,10,15)(17,21,23,20)(18,24,27,29)(19,28,26,31)(22,32,25,30)(33,36,37,44)(34,39,41,45)(35,42,43,48)(38,46,40,47);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,19)(18,25)(20,28)(21,31)(22,27)(23,26)(24,32)(29,30)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,5,6,3)(2,7,9,10)(4,12,13,11)(8,14,16,15)(17,22,23,25)(18,26,27,19)(20,30,21,32)(24,31,29,28)(33,34)(35,38)(36,39)(37,41)(40,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^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, 10388249);