#9625436

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

4262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9532632);