#497569

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#61234	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

814 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fh,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404765);