#475703

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

613 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 613 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 382899);