#9938194

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

4448 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=gh,f^2=h,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, 9845390);