#8334360

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

4262 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8241556);