#10480806

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

3979 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)(3,10)(4,13)(5,8)(6,12)(7,9)(11,15)(14,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(36,44)(39,45)(42,48)(46,47);(1,3)(2,6)(4,14)(5,11)(7,12)(8,16)(9,10)(13,15)(17,19,23,21)(18,24,26,20)(22,27,30,29)(25,31,32,28)(33,34)(35,40)(36,45)(37,41)(38,43)(39,44)(42,46)(47,48);(1,4,7,5)(2,8,9,13)(3,11,12,14)(6,15,10,16)(17,19,23,21)(18,20,26,24)(22,27,30,29)(25,28,32,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,48)(45,47);(1,3)(2,6)(4,11)(5,14)(7,12)(8,15)(9,10)(13,16)(17,20,23,24)(18,19,26,21)(22,28,30,31)(25,27,32,29)(33,36,37,44)(34,39,41,45)(35,42,43,48)(38,46,40,47);(1,5,7,4)(2,8,9,13)(3,11,12,14)(6,16,10,15)(17,21,23,19)(18,24,26,20)(22,27,30,29)(25,28,32,31)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,6)(2,3)(4,15)(5,16)(7,10)(8,11)(9,12)(13,14)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,34)(35,38)(36,39)(37,41)(40,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,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, 10388002);