#406356

order	512
nilpotency class	3
isoclinism	isoclinic class of #395581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1281 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=f,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313552);