#401873

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1202 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^c=fi,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, 309069);