#455068

order	512
nilpotency class	3
isoclinism	isoclinic class of #453996
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1531 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)(9,10,15,30,13,18,14,19)(11,20,24,32,22,16,23,33)(12,25,29,17,27,36,28,35)(21,34,26,37,38,40,39,31);(5,6,8,7)(9,11,13,22)(10,16,18,20)(12,26,27,39)(14,23,15,24)(17,34,35,40)(19,32,30,33)(21,29,38,28)(25,31,36,37);(6,7)(9,12)(10,17)(11,21)(13,27)(14,29)(15,28)(16,31)(18,35)(19,36)(20,37)(22,38)(23,26)(24,39)(25,30)(32,34)(33,40);(2,4)(5,7,8,6)(12,27)(17,35)(21,38)(25,36)(26,39)(28,29)(31,37)(34,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=ei,f^2=i,g^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

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, 362264);