#8943950

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

4229 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)(3,11)(4,7)(5,9)(6,10)(8,14)(12,16)(13,15)(17,18)(19,26)(20,28)(21,25)(22,23)(24,30)(27,31)(29,32)(33,34,39,41)(35,42,36,40)(37,38,48,47)(43,46,45,44);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,10)(15,16)(17,19,23,25)(18,21,22,26)(20,27,30,32)(24,31,28,29)(33,34,39,41)(35,42,36,40)(37,47,48,38)(43,44,45,46);(5,13)(6,12)(9,15)(10,16)(18,22)(21,26)(24,28)(29,31)(33,35,39,36)(34,40,41,42)(37,45,48,43)(38,46,47,44);(1,4,7,2)(3,11,14,8)(5,9,13,15)(6,16,12,10)(17,20)(18,24)(19,27)(21,31)(22,28)(23,30)(25,32)(26,29)(33,36,39,35)(34,40,41,42)(37,43,48,45)(38,46,47,44);(1,5,7,13)(2,9,4,15)(3,12,14,6)(8,10,11,16)(17,21,23,26)(18,25,22,19)(20,29,30,31)(24,27,28,32)(33,37)(34,38)(35,43)(36,45)(39,48)(40,44)(41,47)(42,46);(1,6,7,12)(2,10,4,16)(3,13,14,5)(8,9,11,15)(17,22,23,18)(19,26,25,21)(20,28,30,24)(27,29,32,31)(33,38)(34,37)(35,44)(36,46)(39,47)(40,43)(41,48)(42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8851146);