#454965

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	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1477 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)(9,10,14,19,16,13,15,30)(11,17,22,33,24,21,23,34)(12,25,28,27,29,36,31,18)(20,37,39,38,40,35,26,32);(2,5)(6,8)(9,11)(10,17)(12,26)(13,21)(14,22)(15,23)(16,24)(18,35)(19,33)(20,28)(25,32)(27,37)(29,39)(30,34)(31,40)(36,38);(3,7)(6,8)(9,12)(10,18)(11,20)(13,27)(14,31)(15,28)(16,29)(17,32)(19,36)(21,38)(22,26)(23,39)(24,40)(25,30)(33,35)(34,37);(1,3,4,7)(2,6,5,8)(9,13,16,10)(11,21,24,17)(12,27,29,18)(14,30,15,19)(20,38,40,32)(22,34,23,33)(25,28,36,31)(26,37,39,35)

Polycyclic presentation:

2^9|a^2=f,d^2=ei,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, 362161);