#8772984

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

4244 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)(3,10,9,5)(4,8,12,14)(11,16,15,13)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34,38,42)(35,36,40,39)(37,41,46,48)(43,44,45,47);(2,7)(4,12)(5,10)(11,15)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(33,35,38,40)(34,39,42,36)(37,45,46,43)(41,44,48,47);(2,7)(3,9)(4,12)(13,16)(18,19)(20,27)(24,26)(28,32)(33,34,38,42)(35,39,40,36)(37,41,46,48)(43,47,45,44);(1,3,6,9)(2,5,7,10)(4,11,12,15)(8,13,14,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,45,48,43);(1,5)(2,9)(3,7)(4,13)(6,10)(8,15)(11,14)(12,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,37,38,46)(34,41,42,48)(35,43,40,45)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8680180);