#462007

order	512
nilpotency class	3
isoclinism	isoclinic class of #454977
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

631 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)(4,8)(5,7)(9,10,14,18,13,17,26,34)(11,15,22,32,21,31,27,35)(12,23,25,36,24,16,28,33)(19,37,20,39,38,29,40,30);(1,3,5,4)(2,6,7,8)(9,11,13,21)(10,15,17,31)(12,20,24,40)(14,27,26,22)(16,30,23,39)(18,35,34,32)(19,28,38,25)(29,36,37,33);(1,4,5,3)(2,6,7,8)(9,12,13,24)(10,16,17,23)(11,19,21,38)(14,28,26,25)(15,29,31,37)(18,36,34,33)(20,22,40,27)(30,32,39,35);(9,12)(10,16,17,23)(11,20)(13,24)(14,25)(15,30,31,39)(18,33,34,36)(19,27)(21,40)(22,38)(26,28)(29,32,37,35)

Polycyclic presentation:

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

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, 369203);