#8159430

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

4228 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=g,f^2=g,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, 8066626);