#455137

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1547 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=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, 362333);