#455050

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

1247 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:

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

Polycyclic presentation:

2^9|a^2=ef,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, 362246);