#427259

order	512
nilpotency class	3
isoclinism	isoclinic class of #417622
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1320 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)(3,6)(4,5)(7,8)(9,10,15,20)(11,16,23,32)(12,17,25,35)(13,19,14,18)(21,29,37,40)(22,30,27,31)(24,39,38,33)(26,34,28,36);(9,11)(10,16)(12,24)(13,27)(14,22)(15,23)(17,33)(18,30)(19,31)(20,32)(21,26)(25,38)(28,37)(29,36)(34,40)(35,39);(2,5)(6,8)(9,12)(10,17)(11,21)(13,28)(14,26)(15,25)(16,29)(18,34)(19,36)(20,35)(22,38)(23,37)(24,27)(30,33)(31,39)(32,40);(1,3,4,7)(2,6,5,8)(10,18)(13,14)(16,30)(17,34)(19,20)(22,27)(24,38)(26,28)(29,33)(31,32)(35,36)(39,40)

Polycyclic presentation:

2^9|a^2=i,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^c=gi,g^d=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, 334455);