#451143

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2404
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1630 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358339);