#8720670

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

4097 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=i,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627866);