#8456975

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

4107 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)(18,21)(20,24)(25,29)(28,30)(33,34,39,43)(35,40,46,44)(36,38,42,48)(37,47,45,41);(1,3,8,6)(2,9,7,12)(4,14,5,13)(10,16,11,15)(17,18,23,21)(19,24,27,20)(22,25,31,29)(26,30,32,28)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(1,4,8,5)(2,10,7,11)(3,13,6,14)(9,15,12,16)(17,19,23,27)(18,24,21,20)(22,26,31,32)(25,30,29,28)(33,35,39,46)(34,40,43,44)(36,45,42,37)(38,41,48,47);(1,5,8,4)(2,11,7,10)(3,14,6,13)(9,16,12,15)(17,20)(18,19)(21,27)(22,30)(23,24)(25,32)(26,29)(28,31)(33,36)(34,38)(35,45)(37,46)(39,42)(40,41)(43,48)(44,47);(1,6,8,3)(2,12,7,9)(4,14,5,13)(10,16,11,15)(17,21,23,18)(19,20,27,24)(22,29,31,25)(26,28,32,30)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,7,8,2)(3,12,6,9)(4,11,5,10)(13,16,14,15)(17,22,23,31)(18,25,21,29)(19,26,27,32)(20,28,24,30)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,45)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8364171);