#457026

order	512
nilpotency class	3
isoclinism	isoclinic class of #456417
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1066 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)(3,6)(4,5)(7,8)(9,10,15,19,14,13,16,29)(11,20,23,35,22,17,24,32)(12,25,27,28,26,33,30,18)(21,36,38,31,37,40,39,34);(2,5)(6,8)(9,11,15,23,14,22,16,24)(10,17,19,32,13,20,29,35)(12,21,27,38,26,37,30,39)(18,31,25,40,28,34,33,36);(3,7)(6,8)(9,12,14,26)(10,18,13,28)(11,21,22,37)(15,30,16,27)(17,31,20,34)(19,33,29,25)(23,39,24,38)(32,36,35,40);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,20,22,17)(12,18,26,28)(15,29,16,19)(21,31,37,34)(23,35,24,32)(25,30,33,27)(36,38,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364222);