#10143622

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	#53740	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4050 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=ghi,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, 10050818);