#8943990

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

4225 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)(17,18,23,25)(19,29,32,24)(20,21,27,26)(22,30,31,28)(33,34)(35,41)(36,45)(37,39)(38,42)(40,43)(44,47)(46,48);(1,3)(2,8)(4,11)(5,12)(6,13)(7,14)(9,10)(15,16)(17,19,23,32)(18,24,25,29)(20,30,27,28)(21,22,26,31)(33,35,40,44)(34,41,43,47)(36,46,37,42)(38,45,48,39);(5,13)(6,12)(9,15)(10,16)(18,25)(21,26)(22,31)(24,29)(33,36,40,37)(34,39,43,45)(35,42,44,46)(38,41,48,47);(1,4,7,2)(3,11,14,8)(5,9,13,15)(6,16,12,10)(17,20)(18,26)(19,30)(21,25)(22,29)(23,27)(24,31)(28,32)(33,37,40,36)(34,39,43,45)(35,42,44,46)(38,47,48,41);(1,5,7,13)(2,9,4,15)(3,12,14,6)(8,10,11,16)(17,21,23,26)(18,27,25,20)(19,31,32,22)(24,30,29,28)(33,38)(34,42)(35,39)(36,47)(37,41)(40,48)(43,46)(44,45);(1,6,7,12)(2,10,4,16)(3,13,14,5)(8,9,11,15)(17,22,23,31)(18,28,25,30)(19,26,32,21)(20,29,27,24)(33,39)(34,36)(35,38)(37,43)(40,45)(41,46)(42,47)(44,48)

Polycyclic presentation:

2^9|a^2=hi,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, 8851186);