#10512753

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

4436 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=gh,e^2=hi,f^2=h,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, 10419949);