#10142172

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	#48309	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=ghi,e^2=h,f^2=g,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, 10049368);