#10143606

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	#56909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18,23,25)(19,24,28,22)(20,26,30,31)(21,27,29,32)(33,34)(35,42)(36,41)(37,38)(39,43)(40,47)(44,46)(45,48);(2,7)(4,10)(9,13)(12,14)(17,19,23,28)(18,24,25,22)(20,29,30,21)(26,32,31,27)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43);(2,7)(3,8)(9,13)(11,16)(18,25)(19,28)(20,30)(27,32)(33,36,39,37)(34,41,43,38)(35,44,47,45)(40,48,42,46);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,14)(15,16)(17,20)(18,26)(19,29)(21,28)(22,27)(23,30)(24,32)(25,31)(33,37,39,36)(34,41,43,38)(35,45,47,44)(40,48,42,46);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,21,23,29)(18,27,25,32)(19,30,28,20)(22,26,24,31)(33,38,39,41)(34,37,43,36)(35,46,47,48)(40,45,42,44);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,22,23,24)(18,28,25,19)(20,27,30,32)(21,31,29,26)(33,35)(34,42)(36,44)(37,45)(38,48)(39,47)(40,43)(41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=ghi,e^2=gh,f^2=h,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, 10050802);