#10143319

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	#47913	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,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, 10050515);