#206695

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=di,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113891);