#436711

order	512
nilpotency class	3
isoclinism	isoclinic class of #422233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1156 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:

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

Polycyclic presentation:

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

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, 343907);