#404395

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

1143 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^c=hi,f^d=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, 311591);