#8125489

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

4157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8032685);