#10412231

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

4781 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^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, 10319427);