#8393403

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
prime radical	#49927	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300599);