#10143367

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	#53504	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=i,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10050563);