#117832

order	512
nilpotency class	3
isoclinism	isoclinic class of #117832
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

643 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,f^b=fh,d^c=di,f^c=fhi,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, 25028);