#10482467

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

4476 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)(3,4)(5,7)(6,8)(9,10,15,19)(11,18,22,16)(12,14,13,17)(20,21,23,24)(28,36)(31,37)(34,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,23)(13,20)(14,21)(15,22)(17,24)(18,19)(25,26)(27,32)(28,37)(29,33)(30,35)(31,36)(34,38)(39,40);(9,12,15,13)(10,17,19,14)(11,20,22,23)(16,21,18,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,40)(37,39);(1,2)(3,4)(5,7)(6,8)(9,13,15,12)(10,14,19,17)(11,20,22,23)(16,21,18,24)(25,28,29,36)(26,31,33,37)(27,34,35,40)(30,38,32,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,18)(12,20)(13,23)(14,21)(15,22)(16,19)(17,24)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(1,4,5,6)(2,3,7,8)(9,14)(10,12)(11,21)(13,19)(15,17)(16,23)(18,20)(22,24)(25,26)(27,30)(28,31)(29,33)(32,35)(34,38)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10389663);