#107770

order	512
nilpotency class	3
isoclinism	isoclinic class of #9303
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^c=ei,f^c=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14966);