#392310

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,c^b=ci,h^d=hi

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, 299506);