#8840302

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

4555 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:

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

Polycyclic presentation:

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

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, 8747498);