#455202

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1354 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362398);