#10324024

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

4365 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,e^2=gh,f^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, 10231220);