#8634474

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=gi,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, 8541670);