#10143330

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	#55942	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=gi,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, 10050526);