#8119562

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

3859 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)(20,28)(24,29)(26,32)(30,31)(33,34,39,43)(35,42,38,40)(36,37,41,48)(44,47,46,45);(1,3,8,6)(2,9,7,12)(4,14,5,13)(10,16,11,15)(18,22)(23,25)(24,29)(30,31)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,45,48,47);(1,4,8,5)(2,10,7,11)(3,13,6,14)(9,15,12,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,36)(34,37)(35,44)(38,46)(39,41)(40,45)(42,47)(43,48);(1,5,8,4)(2,11,7,10)(3,14,6,13)(9,16,12,15)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,45,48,47);(1,6,8,3)(2,12,7,9)(4,14,5,13)(10,16,11,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,46)(42,44);(1,7,8,2)(3,12,6,9)(4,11,5,10)(13,16,14,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8026758);