#107094

order	512
nilpotency class	3
isoclinism	isoclinic class of #9383
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1178 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)(9,10,13,17,14,18,25,31)(11,19,22,33,21,15,26,28)(12,16,24,30,23,29,36,39)(20,27,35,38,34,32,37,40);(2,5,7,6)(9,11)(10,15,18,19)(12,20)(13,22)(14,21)(16,27,29,32)(17,28,31,33)(23,34)(24,35)(25,26)(30,38,39,40)(36,37);(9,12)(10,16)(11,20)(13,24)(14,23)(15,27)(17,30)(18,29)(19,32)(21,34)(22,35)(25,36)(26,37)(28,38)(31,39)(33,40)

Polycyclic presentation:

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

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, 14290);