#10143271

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	#54303	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=i,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, 10050467);