#472327

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

922 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)(3,6)(4,5)(7,8)(10,15)(12,23)(13,26)(16,31)(17,33)(18,30)(19,28)(20,24)(21,27)(25,37)(32,35)(36,39);(2,5)(6,8)(9,10,13,15,14,18,26,30)(11,19,27,31,22,16,21,28)(12,17,23,29,25,33,37,40)(20,35,39,34,36,32,24,38);(3,7)(6,8)(9,11)(10,16,18,19)(12,24)(13,27)(14,22)(15,28,30,31)(17,34,33,38)(20,23)(21,26)(25,39)(29,32,40,35)(36,37);(1,3,4,7)(2,6,5,8)(9,12,13,23,14,25,26,37)(10,17,15,29,18,33,30,40)(11,20,21,24,22,36,27,39)(16,32,31,34,19,35,28,38)

Polycyclic presentation:

2^9|b^2=gh,c^2=f,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=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, 379523);