#455142

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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1541 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)(9,10,14,19,15,20,13,18)(11,16,24,21,25,32,23,31)(12,26,28,34,29,33,27,17)(22,36,38,40,39,35,37,30);(5,6,8,7)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,30)(18,31)(19,21)(20,32)(26,36)(27,37)(28,38)(29,39)(33,35)(34,40);(2,4)(6,7)(9,10,15,20)(11,21,25,31)(12,17,29,34)(13,19,14,18)(16,23,32,24)(22,35,39,36)(26,28,33,27)(30,38,40,37);(5,7,8,6)(9,12,15,29)(10,17,20,34)(11,22,25,39)(13,28,14,27)(16,30,32,40)(18,26,19,33)(21,35,31,36)(23,38,24,37)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=i,d^2=ei,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, 362338);