#253916

order	512
nilpotency class	3
isoclinism	isoclinic class of #253891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

938 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)(3,5)(4,7)(6,8)(9,10,14,18)(11,15,21,29)(12,16,22,30)(13,24,26,17)(19,32,25,27)(20,34,35,28)(23,37,36,31)(33,39,38,40);(1,2)(3,5)(4,7)(6,8)(11,19,21,25)(15,27,29,32)(20,33,35,38)(28,39,34,40);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,25)(14,21)(16,28)(17,32)(18,29)(19,26)(22,35)(23,38)(24,27)(30,34)(31,40)(33,36)(37,39);(9,12,14,22)(10,16,18,30)(11,20,21,35)(13,23,26,36)(15,28,29,34)(17,31,24,37)(19,33,25,38)(27,39,32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161112);