#450928

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1126 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358124);