#10504970

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

4050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10412166);