#10349035

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

4263 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=gh,e^2=gh,f^2=h,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, 10256231);