#453476

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

660 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)(9,10,15,20)(11,21,24,26)(12,17)(13,18)(14,32,29,19)(16,34,35,23)(22,27)(25,31)(28,37)(30,40)(33,39)(36,38);(3,7)(5,8)(9,11,14,23,15,24,29,34)(10,16,19,21,20,35,32,26)(12,25,27,40,28,18,38,39)(13,17,30,36,31,37,33,22);(1,3,4,7)(2,5,6,8)(9,12,15,28)(10,17,20,37)(11,18,24,25)(13,26,31,21)(14,27,29,38)(16,33,35,30)(19,36,32,22)(23,39,34,40);(9,13,29,30,15,31,14,33)(10,18,19,40,20,25,32,39)(11,22,34,17,24,36,23,37)(12,26,38,16,28,21,27,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 360672);