#118319

order	512
nilpotency class	3
isoclinism	isoclinic class of #9946
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

737 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343055, #72343059, #72343078, …
maximal normal subgroup of	#72343055, #72343059, #72343078, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25515);