#8603529

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	#35053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 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:

(2,6)(7,9)(8,12)(14,15)(17,18,23,25)(19,21,22,24)(20,28,31,26)(27,32,29,30)(34,37)(35,36)(42,45)(43,44);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,23,22)(18,24,25,21)(20,29,31,27)(26,30,28,32)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(3,10)(7,9)(11,16)(14,15)(18,25)(21,24)(26,28)(30,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,20,23,31)(18,26,25,28)(19,27,22,29)(21,30,24,32)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,21,23,24)(18,19,25,22)(20,30,31,32)(26,27,28,29)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,19)(18,24,25,21)(20,27,31,29)(26,30,28,32)(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=h,c^2=i,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,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, 8510725);