#345916

order	512
nilpotency class	3
isoclinism	isoclinic class of #345888
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

513 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)(11,21)(12,27)(13,19)(14,18)(15,20)(16,24)(17,29)(22,37)(23,26)(25,33)(28,38)(30,35)(31,36)(32,40)(34,39);(2,5)(3,6)(7,8)(9,11,14,25,15,26,13,24)(10,16,19,23,20,33,18,21)(12,28,29,22,31,39,30,40)(17,34,27,37,35,38,36,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,22)(13,29)(14,30)(15,31)(16,32)(18,27)(19,36)(20,35)(21,34)(23,38)(24,39)(25,28)(26,40)(33,37);(1,2,4,5)(3,6,7,8)(9,10)(11,23)(12,17)(13,19)(14,18)(15,20)(16,25)(21,26)(22,38)(24,33)(27,30)(28,32)(29,36)(31,35)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=defi,e^a=eh,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,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, 253112);