#10480922

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

4269 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)(4,7,12,9)(5,8,14,10)(11,15,13,16)(18,19)(21,29)(23,25)(27,30)(33,34,40,38)(35,37,41,36)(39,46,47,42)(43,44,48,45);(2,3)(7,9)(8,10)(15,16)(18,19)(23,25)(24,26)(31,32)(33,35)(34,36)(37,38)(39,43)(40,41)(42,44)(45,46)(47,48);(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,31,30,32)(33,36,40,37)(34,41,38,35)(39,44,47,45)(42,48,46,43);(1,3,6,2)(4,9,12,7)(5,10,14,8)(11,16,13,15)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,31,30,32)(33,37,40,36)(34,41,38,35)(39,45,47,44)(42,48,46,43);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,23)(19,25)(21,30)(22,28)(24,32)(26,31)(27,29)(33,38,40,34)(35,37,41,36)(39,42,47,46)(43,44,48,45);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,31)(25,32)(28,30)(33,39,40,47)(34,42,38,46)(35,43,41,48)(36,44,37,45)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gi,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, 10388118);