#480084

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,e^2=i,f^2=h,h^2=i,b^a=be,d^a=di,e^a=eg,c^b=ci,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, 387280);