#474235

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

633 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)(9,10,14,19)(11,13)(12,24,26,17)(15,20,30,33)(16,22)(18,32)(21,25)(23,29)(27,36,31,35)(28,37)(34,39)(38,40);(9,11,15,23,14,22,30,18)(10,16,33,29,19,13,20,32)(12,21,27,34,26,37,31,38)(17,25,36,40,24,28,35,39);(2,5)(6,8)(9,12,14,26)(10,17,19,24)(11,21,22,37)(13,28,16,25)(15,31,30,27)(18,34,23,38)(20,36,33,35)(29,39,32,40);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,18,19,23)(11,20,22,33)(12,25,26,28)(15,32,30,29)(17,34,24,38)(21,36,37,35)(27,40,31,39)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,d^c=de,h^c=hi,g^d=gi,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, 381431);