#8720372

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=hi,e^2=g,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, 8627568);