#10183154

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	#44249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 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)(3,6)(4,9)(5,10)(7,8)(11,16)(12,14)(13,15)(17,18,22,24)(19,20,27,23)(21,25,31,32)(26,28,29,30)(33,34,40,44)(35,41,38,43)(36,42,37,39)(45,46,47,48);(1,3,7,11)(2,6,8,16)(4,12,13,5)(9,14,15,10)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,35,40,38)(34,41,44,43)(36,47,37,45)(39,46,42,48);(2,8)(3,11)(4,13)(10,14)(18,24)(20,23)(25,32)(28,30)(33,36,40,37)(34,42,44,39)(35,45,38,47)(41,46,43,48);(1,4)(2,9)(3,12)(5,11)(6,14)(7,13)(8,15)(10,16)(17,20,22,23)(18,19,24,27)(21,30,31,28)(25,29,32,26)(33,37,40,36)(34,39,44,42)(35,45,38,47)(41,46,43,48);(1,5,7,12)(2,10,8,14)(3,13,11,4)(6,15,16,9)(17,21,22,31)(18,25,24,32)(19,26,27,29)(20,28,23,30)(33,38,40,35)(34,43,44,41)(36,47,37,45)(39,46,42,48);(1,6)(2,11)(3,8)(4,14)(5,9)(7,16)(10,13)(12,15)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,39)(34,36)(35,46)(37,44)(38,48)(40,42)(41,47)(43,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=hi,e^2=gi,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, 10090350);