#128000

order	512
nilpotency class	4
isoclinism	isoclinic class of #7304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#342
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂²	C₂×#342
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1412 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1412 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,f^2=i,b^a=bd,d^a=dg,f^a=fh,g^a=gi,d^b=dh,e^b=eg,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267929
maximal normal subgroup of	#72267929
maximal quotient of	#72268051

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35196);