#128055

order	512
nilpotency class	4
isoclinism	isoclinic class of #128055
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

1408 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267925, #72267929
maximal normal subgroup of	#72267925, #72267929
maximal quotient of	#72268051, #72268052

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35251);