#411123

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
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

683 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318319);