#431974

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1140 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ef,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 339170);