#8228384

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

4094 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,15)(14,16)(17,18,22,24)(19,23,27,32)(20,25,29,30)(21,28,31,26);(2,7)(3,10)(4,13)(14,15)(17,19)(18,23)(20,28)(21,30)(22,27)(24,32)(25,31)(26,29)(33,34,38,39)(35,37,41,46)(36,43,45,40)(42,48,47,44);(2,7)(5,8)(9,12)(14,15)(18,24)(19,27)(25,30)(26,28)(34,39)(37,46)(40,43)(44,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,20)(18,25)(19,26)(21,23)(22,29)(24,30)(27,28)(31,32)(33,35,38,41)(34,37,39,46)(36,42,45,47)(40,44,43,48);(1,4,6,13)(2,9,7,12)(3,11,10,16)(5,14,8,15)(17,21)(18,26)(19,25)(20,23)(22,31)(24,28)(27,30)(29,32)(33,36,38,45)(34,40,39,43)(35,42,41,47)(37,44,46,48);(1,5,6,8)(2,3,7,10)(4,14,13,15)(9,11,12,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,32)(28,29)(30,31)(33,37)(34,41)(35,39)(36,44)(38,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8135580);