#10171717

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4066 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=ghi,e^2=gi,f^2=g,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, 10078913);