#8712773

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

4098 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)(3,8)(4,12)(5,9)(6,10)(7,13)(11,15)(14,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(34,37)(35,36)(42,45)(43,44);(1,3)(2,5)(4,11)(6,9)(7,15)(8,10)(12,14)(13,16)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(4,13)(7,12)(11,16)(14,15)(19,28)(20,29)(24,30)(25,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,4)(2,7)(3,11)(5,15)(6,13)(8,14)(9,16)(10,12)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,3)(2,8)(4,11)(5,10)(6,9)(7,14)(12,15)(13,16)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,5,6,8)(2,9,10,3)(4,14,13,15)(7,11,12,16)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=h,e^2=h,f^2=g,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, 8619969);