#346223

order	512
nilpotency class	3
isoclinism	isoclinic class of #346096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

509 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=efi,c^2=ehi,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=defi,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253419);