#8196023

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

3998 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=h,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, 8103219);