#9936412

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

4678 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)(5,8)(6,7)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,30,33)(27,29,35,31)(28,32,37,40)(34,38,39,36);(1,3)(2,4)(5,6)(7,8)(12,19)(15,17)(20,24)(22,23)(25,27,30,35)(26,31,33,29)(28,34,37,39)(32,36,40,38);(1,4)(2,6)(3,8)(5,7)(10,11)(15,17)(16,18)(22,23)(25,27,30,35)(26,31,33,29)(28,34,37,39)(32,36,40,38);(1,2,5,8)(3,7,6,4)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27)(26,29)(28,34)(30,35)(31,33)(32,38)(36,40)(37,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28)(26,32)(27,34)(29,38)(30,37)(31,36)(33,40)(35,39);(1,3)(2,7)(4,8)(5,6)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,27)(28,36)(30,31)(32,39)(33,35)(34,40)(37,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=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, 9843608);