#474250

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2572
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

803 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)(6,8)(9,10,14,17,13,18,29,19)(11,20,23,32,22,15,30,33)(12,24,27,36,26,16,28,35)(21,31,25,34,38,37,39,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,25,26,39)(13,22)(14,23)(16,34,24,40)(17,33)(18,20)(19,32)(21,27,38,28)(29,30)(31,35,37,36);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,21)(13,26)(14,28)(15,31)(17,36)(18,24)(19,35)(20,37)(22,38)(23,39)(25,30)(27,29)(32,34)(33,40);(3,7)(6,8)(10,17,18,19)(11,22)(12,26)(15,32,20,33)(16,35,24,36)(23,30)(27,28)(31,40,37,34)

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=de,g^c=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, 381446);