#10086353

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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)(17,18,22,23)(19,25,26,24)(20,21,30,29)(27,31,32,28)(33,34,39,43)(35,41,46,37)(36,40,38,42)(44,47,45,48);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,18,22,23)(19,25,26,24)(20,29,30,21)(27,28,32,31)(33,35,39,46)(34,37,43,41)(36,44,38,45)(40,48,42,47);(2,7)(4,10)(9,13)(12,14)(18,23)(19,26)(20,30)(28,31)(33,36,39,38)(34,40,43,42)(35,44,46,45)(37,48,41,47);(1,2)(3,10)(4,8)(5,9)(6,7)(11,14)(12,16)(13,15)(17,19,22,26)(18,24,23,25)(20,27,30,32)(21,28,29,31)(33,35)(34,41)(36,44)(37,43)(38,45)(39,46)(40,47)(42,48);(1,4,6,10)(2,8,7,3)(5,14,15,12)(9,11,13,16)(17,20,22,30)(18,21,23,29)(19,27,26,32)(24,28,25,31)(33,37,39,41)(34,35,43,46)(36,47,38,48)(40,45,42,44);(1,5)(2,9)(3,11)(4,12)(6,15)(7,13)(8,16)(10,14)(17,21)(18,20)(19,28)(22,29)(23,30)(24,27)(25,32)(26,31)(33,38,39,36)(34,42,43,40)(35,45,46,44)(37,47,41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9993549);