#10444934

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

4457 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10352130);