#405577

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#23406	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1340 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)(3,6)(4,5)(7,8)(9,10)(11,23)(12,28)(13,32)(14,22)(15,20)(16,21)(17,25)(18,30)(19,24)(26,37)(27,36)(29,39)(31,38)(33,35)(34,40);(2,5)(6,8)(9,11,15,26,16,27,14,25)(10,17,22,36,21,37,20,23)(12,24,30,34,31,33,29,13)(18,19,28,32,39,35,38,40);(9,12)(10,18)(11,24)(13,25)(14,29)(15,30)(16,31)(17,35)(19,37)(20,38)(21,39)(22,28)(23,32)(26,34)(27,33)(36,40);(1,3,4,7)(2,6,5,8)(9,13,14,33,16,34,15,24)(10,19,20,40,21,35,22,32)(11,12,25,29,27,31,26,30)(17,28,23,18,37,38,36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312773);