#449633

order	512
nilpotency class	3
isoclinism	isoclinic class of #417534
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1328 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:

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

Polycyclic presentation:

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

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, 356829);