#8196392

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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)(3,9)(8,11)(10,15)(18,23)(19,20)(26,29)(27,28)(33,34,38,41)(35,36,39,43)(37,40,46,48)(42,44,45,47);(2,6)(7,13)(8,11)(14,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(35,39)(36,43)(42,45)(44,47);(1,2,5,6)(3,7,9,13)(4,11,12,8)(10,16,15,14)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,32)(28,31)(33,35,38,39)(34,36,41,43)(37,45,46,42)(40,47,48,44);(17,19,22,20)(18,24,23,25)(21,27,30,28)(26,31,29,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,45)(42,48)(46,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,19)(18,25,23,24)(21,28,30,27)(26,32,29,31);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21,22,30)(18,26,23,29)(19,27,20,28)(24,31,25,32)(33,37,38,46)(34,40,41,48)(35,42,39,45)(36,44,43,47)

Polycyclic presentation:

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

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, 8103588);