#496548

order	512
nilpotency class	4
isoclinism	isoclinic class of #480684
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

899 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)(3,7)(9,10,14,18,12,15,24,31)(11,19,22,37,20,34,28,39)(13,25,27,36,23,17,29,35)(16,32,33,26,30,38,40,21);(1,2)(3,6)(4,5)(7,8)(10,15)(11,20)(17,25)(18,31)(21,26)(22,28)(32,38)(35,36);(2,5)(6,8)(9,11,12,20)(10,16,15,30)(13,26,23,21)(14,28,24,22)(17,34,25,19)(18,33,31,40)(27,32,29,38)(35,39,36,37);(1,3,4,7)(2,6,5,8)(10,15)(11,20)(17,25)(18,31)(21,26)(22,28)(32,38)(35,36)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,f^c=fi,g^c=ghi,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, 403744);