#10506327

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

4202 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,14)(15,16)(17,18)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(33,34,40,38)(35,37,41,36)(39,46,47,42)(43,44,48,45);(2,7)(3,11)(9,12)(10,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35)(34,36)(37,38)(39,43)(40,41)(42,44)(45,46)(47,48);(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(33,36,40,37)(34,41,38,35)(39,44,47,45)(42,48,46,43);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,12,15,9)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,37,40,36)(34,41,38,35)(39,45,47,44)(42,48,46,43);(1,4)(2,9)(3,10)(5,8)(6,14)(7,12)(11,16)(13,15)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,38,40,34)(35,37,41,36)(39,42,47,46)(43,44,48,45);(1,5)(2,10)(3,12)(4,13)(6,15)(7,16)(8,14)(9,11)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,39,40,47)(34,42,38,46)(35,43,41,48)(36,44,37,45)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=g,e^2=hi,f^2=h,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, 10413523);