#9938093

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

4312 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=gh,e^2=g,f^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, 9845289);