#398492

order	512
nilpotency class	3
isoclinism	isoclinic class of #394358
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1105 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=f,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305688);