#427104

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

847 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=ef,d^2=e,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, 334300);