#118269

order	512
nilpotency class	3
isoclinism	isoclinic class of #118267
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,d^c=di,e^c=eh,g^c=gh

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, 25465);