#346960

order	512
nilpotency class	3
isoclinism	isoclinic class of #346950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 254156);