#345980

order	512
nilpotency class	3
isoclinism	isoclinic class of #345968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#28243	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

529 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:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253176);