#10412328

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

4672 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,e^2=g,f^2=g,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, 10319524);