#480086

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂×#525	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,e^2=i,f^2=h,h^2=i,b^a=be,c^a=ci,e^a=eg,d^b=di,e^b=ei,f^b=fgi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387282);