#484096

order	512
nilpotency class	4
isoclinism	isoclinic class of #484095
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#508	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

439 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286241, #72286279, #72286506, …
maximal normal subgroup of	#72286241, #72286279, #72286506, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391292);