#496731

order	512
nilpotency class	4
isoclinism	isoclinic class of #483439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

554 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)(9,10,11,20,13,18,15,16)(12,23,24,38,25,37,21,36)(14,28,22,35,27,31,29,19)(17,32,30,26,34,39,33,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,16,18,20)(12,24,25,21)(14,22,27,29)(17,33,34,30)(19,31,35,28)(23,36,37,38)(26,39,40,32);(2,5)(6,8)(9,12,14,26,15,24,29,39,13,25,27,40,11,21,22,32)(10,17,19,23,20,33,28,36,18,34,35,37,16,30,31,38);(1,3,4,7)(2,6,5,8)(10,18)(16,20)(17,34)(19,35)(23,37)(28,31)(30,33)(36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403927);