#8159011

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

4049 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)(3,4)(5,7)(6,13)(8,10)(9,14)(11,16)(12,15)(18,20)(22,30)(24,25)(27,29)(33,34)(35,41)(36,43)(37,45)(38,40)(39,46)(42,44)(47,48);(1,3,7,8)(2,4,5,10)(6,11,14,15)(9,12,13,16)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34,38,40)(35,41,43,36)(37,39,46,45)(42,47,48,44);(1,4,7,10)(2,8,5,3)(6,12,14,16)(9,15,13,11)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,34,38,40)(35,41,43,36)(37,39,46,45)(42,47,48,44);(1,4)(2,3)(5,8)(6,12)(7,10)(9,11)(13,15)(14,16)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,35,38,43)(34,36,40,41)(37,42,46,48)(39,44,45,47);(1,5,7,2)(3,10,8,4)(6,13,14,9)(11,16,15,12)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,6,7,14)(2,9,5,13)(3,11,8,15)(4,12,10,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,37,38,46)(34,39,40,45)(35,42,43,48)(36,44,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066207);