#457878

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
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

857 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 365074);