#481111

order	512
nilpotency class	3
isoclinism	isoclinic class of #17547
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	#3475	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

773 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=gi,f^2=h,h^2=i,b^a=be,e^a=eg,e^b=egi,f^b=fi,d^c=di

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388307);