#8930529

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

4353 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8837725);