#428397

order	512
nilpotency class	3
isoclinism	isoclinic class of #417728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1063 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:

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

Polycyclic presentation:

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

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, 335593);