#181713

order	512
nilpotency class	3
isoclinism	isoclinic class of #17547
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1546 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)(3,5)(4,7)(6,8)(9,10,13,18,15,19,14,28)(11,20,23,34,22,16,24,30)(12,17,26,32,25,31,27,38)(21,33,36,40,35,29,37,39);(2,5,7,6)(9,11,14,24,15,22,13,23)(10,16,18,30,19,20,28,34)(12,21,27,37,25,35,26,36)(17,29,32,39,31,33,38,40);(9,12)(10,17)(11,21)(13,26)(14,27)(15,25)(16,29)(18,32)(19,31)(20,33)(22,35)(23,36)(24,37)(28,38)(30,39)(34,40);(12,25)(17,31)(21,35)(26,27)(29,33)(32,38)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,g^2=i,b^a=be,e^a=eh,g^a=ghi,e^b=ei,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, 88909);