#10143719

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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)(17,18,23,25)(19,20,28,24)(21,26,32,30)(22,29,31,27)(33,34,40,43)(35,44,47,41)(36,38,37,48)(39,42,45,46);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,19,23,28)(18,24,25,20)(21,31,32,22)(26,29,30,27)(33,35)(34,41)(36,39)(37,45)(38,46)(40,47)(42,48)(43,44);(2,8)(3,11)(9,16)(10,12)(18,25)(20,24)(26,30)(27,29)(33,36,40,37)(34,38,43,48)(35,39,47,45)(41,46,44,42);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,20)(18,19)(21,27)(22,30)(23,24)(25,28)(26,31)(29,32)(33,37,40,36)(34,38,43,48)(35,45,47,39)(41,46,44,42);(1,4,6,14)(2,9,8,16)(3,12,11,10)(5,7,15,13)(17,21,23,32)(18,26,25,30)(19,22,28,31)(20,29,24,27)(33,38)(34,37)(35,46)(36,43)(39,44)(40,48)(41,45)(42,47);(1,5,6,15)(2,10,8,12)(3,9,11,16)(4,13,14,7)(17,22)(18,27)(19,21)(20,30)(23,31)(24,26)(25,29)(28,32)(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=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, 10050915);