#456577

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23390	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1424 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi,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, 363773);