#419314

order	512
nilpotency class	3
isoclinism	isoclinic class of #417704
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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326510);