#93946

order	512
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#354
commutator	C₂×C₄²	C₂×C₈
Frattini, cosocle	#1728	C₂²
socle	C₂²	#1324
prime radical	C₂⁴	#101
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1728 ◁ G
LECS	G ▷ #1728 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

216 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 64

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=defhi,d^c=dhi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1142);