#8196929

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=ghi,e^2=h,f^2=ghi,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, 8104125);