#10368620

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

4109 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)(4,5)(6,7)(8,11)(9,14)(12,16)(13,15)(17,18,22,24)(19,25,27,21)(20,23,28,31)(26,32,29,30)(33,34,39,36)(35,37,40,42)(38,41,47,44)(43,45,46,48);(2,7)(3,11)(5,14)(12,15)(17,19,22,27)(18,21,24,25)(20,26,28,29)(23,30,31,32)(33,35,39,40)(34,37,36,42)(38,46,47,43)(41,48,44,45);(2,7)(4,9)(8,10)(12,15)(20,28)(23,31)(26,29)(30,32)(33,34,39,36)(35,42,40,37)(38,41,47,44)(43,48,46,45);(1,3,6,11)(2,8,7,10)(4,12,9,15)(5,13,14,16)(17,19)(18,21)(20,29)(22,27)(23,32)(24,25)(26,28)(30,31)(33,36,39,34)(35,37,40,42)(38,44,47,41)(43,45,46,48);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,20,22,28)(18,23,24,31)(19,26,27,29)(21,30,25,32)(33,37)(34,40)(35,36)(38,45)(39,42)(41,46)(43,44)(47,48);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,21)(18,19)(20,30)(22,25)(23,26)(24,27)(28,32)(29,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10275816);