#108206

order	512
nilpotency class	3
isoclinism	isoclinic class of #9467
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	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1110 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)(9,10,13,18,15,20,14,19)(11,16,23,30,22,29,24,31)(12,17,26,33,25,32,27,34)(21,35,37,40,36,28,38,39);(2,5,7,6)(9,11,14,24,15,22,13,23)(10,16,19,31,20,29,18,30)(12,21,27,38,25,36,26,37)(17,28,34,40,32,35,33,39);(9,12)(10,17)(11,21)(13,26)(14,27)(15,25)(16,28)(18,33)(19,34)(20,32)(22,36)(23,37)(24,38)(29,35)(30,39)(31,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,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, 15402);