#10196259

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	#38713	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,d^2=gh,e^2=g,f^2=i,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, 10103455);