#450922

order	512
nilpotency class	3
isoclinism	isoclinic class of #417562
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1153 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^b=hi,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, 358118);