#212755

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	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

565 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

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,e^a=eh,f^a=fh,c^b=ci,d^b=di,d^c=dh

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, 119951);