#454895

order	512
nilpotency class	3
isoclinism	isoclinic class of #454110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1080 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=i,f^2=i,g^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,g^c=gi,h^c=hi,f^d=fi

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, 362091);