#8726714

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

4208 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 8633910);