#8595729

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	#56394	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4251 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8502925);