#445119

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

964 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:

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

Polycyclic presentation:

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

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, 352315);