#10143724

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3899 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=hi,e^2=gh,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, 10050920);