#10325228

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

4552 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)(4,5)(6,8)(9,10)(11,18)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24)(25,26,31,33)(27,34,28,32)(29,30,39,40)(35,38,37,36);(2,5)(6,7)(11,12)(13,22)(15,18)(16,21);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(25,27,31,28)(26,32,33,34)(29,35,39,37)(30,36,40,38);(1,3,4,8)(2,6,5,7)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,28,31,27)(26,32,33,34)(29,35,39,37)(30,38,40,36);(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,29)(26,30)(27,35)(28,37)(31,39)(32,36)(33,40)(34,38);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,30)(26,29)(27,36)(28,38)(31,40)(32,35)(33,39)(34,37)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10232424);