#10412421

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

4680 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=h,e^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, 10319617);