#408173

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

709 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)(3,6)(4,8)(5,7)(9,10,15,19)(11,20,24,31)(12,25,28,26)(13,29,18,14)(16,23,32,22)(17,33,34,27)(21,30,38,40)(35,36,39,37);(1,3,5,4)(2,6,7,8)(9,11,15,24)(10,16,19,32)(12,21,28,38)(13,20,18,31)(14,23,29,22)(17,30,34,40)(25,39,26,35)(27,37,33,36);(2,7)(6,8)(9,12,14,27,15,28,29,33)(10,17,18,25,19,34,13,26)(11,21,22,36,24,38,23,37)(16,30,20,35,32,40,31,39);(1,4,5,3)(2,8,7,6)(9,13,14,10,15,18,29,19)(11,20,23,16,24,31,22,32)(12,26,27,17,28,25,33,34)(21,35,37,30,38,39,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315369);