#117115

order	512
nilpotency class	3
isoclinism	isoclinic class of #10218
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

754 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)(3,5)(4,7)(6,8)(9,10,14,18,15,20,13,19)(11,21,25,31,24,32,23,16)(12,17,27,35,28,36,29,34)(22,37,26,30,38,33,39,40);(9,11,13,23,15,24,14,25)(10,16,18,21,20,31,19,32)(12,26,27,38,28,39,29,22)(17,33,34,30,36,37,35,40);(2,5)(3,8)(6,7)(9,12)(10,17)(11,22)(13,29)(14,27)(15,28)(16,30)(18,35)(19,34)(20,36)(21,33)(23,39)(24,38)(25,26)(31,40)(32,37)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^b=eh,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24311);