#212794

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
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 ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

534 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)(9,10,14,16)(11,17,19,18)(12,13,21,22)(15,23,24,20)(25,26,30,28,29,32,38,37)(27,31,35,36,33,39,40,34);(2,5)(6,8)(9,11,13,20,14,19,22,23)(10,15,12,17,16,24,21,18)(25,27,29,33)(26,31,32,39)(28,36,37,34)(30,35,38,40);(1,3,4,7)(2,6,5,8)(9,12,13,16,14,21,22,10)(11,18,20,15,19,17,23,24)(26,32)(27,33)(28,37)(35,40);(11,19)(15,24)(17,18)(20,23)(25,28,30,32,29,37,38,26)(27,34,35,31,33,36,40,39)

Polycyclic presentation:

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

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, 119990);