#117753

order	512
nilpotency class	3
isoclinism	isoclinic class of #117753
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

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

659 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)(11,19)(12,23,25,16)(13,26)(14,18)(15,21)(17,27)(20,33,35,31)(22,32)(24,30,39,38)(28,34,37,40)(29,36);(2,5,6,8)(9,11,13,21,14,22,27,36)(10,15,17,19,18,29,26,32)(12,20,24,34,25,35,39,40)(16,28,30,31,23,37,38,33);(1,2)(3,8)(4,6)(5,7)(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,30)(18,23)(19,31)(21,34)(22,35)(26,38)(27,39)(29,37)(32,33)(36,40)

Polycyclic presentation:

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

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, 24949);