#118187

order	512
nilpotency class	3
isoclinism	isoclinic class of #118183
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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

706 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25383);