#454966

order	512
nilpotency class	3
isoclinism	isoclinic class of #454039
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1437 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)(9,10,15,19,14,13,16,30)(11,20,23,34,22,17,24,33)(12,25,28,29,27,36,31,18)(21,35,39,32,38,40,26,37);(2,5)(6,8)(9,11)(10,17)(12,26)(13,20)(14,22)(15,23)(16,24)(18,35)(19,33)(21,28)(25,32)(27,39)(29,40)(30,34)(31,38)(36,37);(3,7)(6,8)(9,12,14,27)(10,18,13,29)(11,21,22,38)(15,31,16,28)(17,32,20,37)(19,36,30,25)(23,26,24,39)(33,35,34,40);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,20,22,17)(12,18,27,29)(15,30,16,19)(21,32,38,37)(23,34,24,33)(25,31,36,28)(26,35,39,40)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,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, 362162);