#10159203

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(3,5)(4,8)(6,10)(7,14)(9,15)(11,16)(12,13)(17,18,24,26)(19,27,23,25)(20,22,21,31)(28,32,29,30)(33,34)(35,45)(36,39)(37,46)(38,42)(40,44)(41,48)(43,47);(1,3,6,9)(2,5,10,15)(4,12,14,11)(7,16,8,13)(17,19,24,23)(18,25,26,27)(20,28,21,29)(22,30,31,32)(33,35)(34,41)(36,46)(37,42)(38,43)(39,47)(40,48)(44,45);(1,4,6,14)(2,7,10,8)(3,11,9,12)(5,13,15,16)(17,20,24,21)(18,22,26,31)(19,28,23,29)(25,30,27,32)(33,36,40,38)(34,42,44,39)(35,46,48,43)(37,45,47,41);(1,4)(2,8)(3,12)(5,13)(6,14)(7,10)(9,11)(15,16)(17,21,24,20)(18,22,26,31)(19,29,23,28)(25,30,27,32)(33,37,40,47)(34,43,44,46)(35,42,48,39)(36,41,38,45);(1,3)(2,5)(4,11)(6,9)(7,13)(8,16)(10,15)(12,14)(17,22)(18,21)(19,30)(20,26)(23,32)(24,31)(25,29)(27,28)(33,38,40,36)(34,42,44,39)(35,46,48,43)(37,41,47,45);(1,5)(2,9)(3,10)(4,13)(6,15)(7,12)(8,11)(14,16)(17,23,24,19)(18,25,26,27)(20,29,21,28)(22,30,31,32)(33,39,40,42)(34,38,44,36)(35,47,48,37)(41,43,45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=g,e^2=hi,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, 10066399);