#496550

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

915 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)(9,10,14,19,15,20,11,16)(12,23,21,38,24,36,29,40)(13,26,22,37,28,18,30,32)(17,33,35,27,34,39,31,25);(1,2)(3,6)(4,5)(7,8)(9,11,15,14)(10,16,20,19)(12,21,24,29)(13,22,28,30)(17,31,34,35)(18,32,26,37)(23,38,36,40)(25,39,27,33);(2,5)(6,8)(9,12,15,24)(10,17,20,34)(11,21,14,29)(13,27,28,25)(16,31,19,35)(18,36,26,23)(22,33,30,39)(32,40,37,38);(1,3,4,7)(2,6,5,8)(12,24)(17,34)(21,29)(23,36)(25,27)(31,35)(33,39)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=e,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, 403746);