#449595

order	512
nilpotency class	3
isoclinism	isoclinic class of #417700
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1960 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:

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

Polycyclic presentation:

2^9|a^2=eh,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,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, 356791);