#8634745

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	#53103	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4203 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=h,e^2=gh,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, 8541941);