#8726694

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	#53116	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4199 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=gi,e^2=i,f^2=gi,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, 8633890);