#460080

order	512
nilpotency class	3
isoclinism	isoclinic class of #454369
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1207 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)(3,7,8,6)(9,10,15,19)(11,20,18,14)(12,16,25,32)(13,26,29,17)(21,24,35,30)(22,28,34,36)(23,31,38,37)(27,39,40,33);(2,5)(6,7)(9,11,14,10,15,18,20,19)(12,21,24,32,25,35,30,16)(13,22,28,17,29,34,36,26)(23,33,27,37,38,39,40,31);(1,3,4,8)(2,6,5,7)(9,12)(10,16)(11,21)(13,27)(14,30)(15,25)(17,33)(18,35)(19,32)(20,24)(22,37)(23,28)(26,39)(29,40)(31,34)(36,38);(3,8)(6,7)(9,13,14,28,15,29,20,36)(10,17,18,34,19,26,11,22)(12,23,24,27,25,38,30,40)(16,31,21,33,32,37,35,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=f,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^c=hi,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, 367276);