#10480918

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

3994 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=i,e^2=g,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, 10388114);