#8845507

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

4268 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8752703);