#128002

order	512
nilpotency class	4
isoclinism	isoclinic class of #128001
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1384 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1384 conjugacy classes of subgroups — too many to draw. Only its 146 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)(16,23,31,38)(22,35)(24,36)(25,39)(26,34,29,33)(30,37,32,40);(1,3)(2,5,6,7)(4,8)(9,11,14,22,15,24,12,25)(10,16,19,30,20,31,17,32)(13,23,29,37,28,38,26,40)(18,21,34,35,27,36,33,39);(9,12,15,14)(10,17,20,19)(11,22,24,25)(13,26,28,29)(16,30,31,32)(18,33,27,34)(21,35,36,39)(23,37,38,40)

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,c^b=ci,d^b=dh,e^b=eg,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267925
maximal normal subgroup of	#72267925
maximal quotient of	#72268051

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35198);