#496563

order	512
nilpotency class	4
isoclinism	isoclinic class of #480670
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

989 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)(3,7)(9,10,14,18,13,17,27,34)(11,19,22,36,21,15,28,30)(12,23,26,35,25,16,31,33)(20,29,37,39,24,32,38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,24)(13,21)(14,22)(16,32)(17,19)(18,30)(20,25)(23,29)(26,38)(27,28)(31,37)(33,40)(34,36)(35,39);(2,5)(6,8)(9,12,13,25)(10,16,17,23)(11,20,21,24)(14,26,27,31)(15,29,19,32)(18,35,34,33)(22,37,28,38)(30,40,36,39);(1,3,4,7)(2,6,5,8)(10,17)(11,21)(16,23)(18,34)(20,24)(22,28)(33,35)(37,38)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,g^c=gh,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403759);