#265179

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1024 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

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, 172375);