#10143718

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	#40701	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

(1,2,6,8)(3,7,11,13)(4,9,14,16)(5,12,15,10)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(33,34)(35,44)(36,41)(37,47)(38,42)(39,43)(40,46)(45,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,19,23,28)(18,24,25,27)(20,26,30,32)(21,29,31,22)(33,35)(34,40)(36,38)(37,45)(39,46)(41,48)(42,47)(43,44);(2,8)(3,11)(9,16)(10,12)(18,25)(20,30)(24,27)(26,32)(33,36,39,37)(34,41,43,47)(35,38,46,45)(40,48,44,42);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,20)(18,22)(19,26)(21,24)(23,30)(25,29)(27,31)(28,32)(33,37,39,36)(34,41,43,47)(35,45,46,38)(40,48,44,42);(1,4,6,14)(2,9,8,16)(3,12,11,10)(5,7,15,13)(17,21,23,31)(18,26,25,32)(19,22,28,29)(20,27,30,24)(33,34)(35,40)(36,41)(37,47)(38,48)(39,43)(42,45)(44,46);(1,5,6,15)(2,10,8,12)(3,9,11,16)(4,13,14,7)(17,22)(18,20)(19,21)(23,29)(24,26)(25,30)(27,32)(28,31)(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=hi,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, 10050914);