#455115

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1499 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)(5,6)(7,8)(9,10,15,18)(11,14,21,27)(12,22,25,16)(13,20,29,37)(17,33,35,28)(19,24,32,30)(23,36,39,34)(26,38,40,31);(9,11,14,18,15,21,27,10)(12,19,24,16,25,32,30,22)(13,20,28,35,29,37,33,17)(23,36,26,31,39,34,40,38);(2,4)(9,12,15,25)(10,16,18,22)(11,19,21,32)(13,26,29,40)(14,30,27,24)(17,34,35,36)(20,31,37,38)(23,33,39,28);(6,8)(9,13,27,33,15,29,14,28)(10,17,21,37,18,35,11,20)(12,23,30,40,25,39,24,26)(16,31,19,36,22,38,32,34)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^c=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, 362311);