#10504962

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

4436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,e^2=gi,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, 10412158);