#128082

order	512
nilpotency class	4
isoclinism	isoclinic class of #128055
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂⁴	C₂×C₄²
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

837 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,b^a=bd,c^a=cg,d^a=dg,f^a=fh,d^b=dh,e^b=eg,h^b=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236729, #72236736, #72238646, …
maximal normal subgroup of	#72236729, #72236736, #72238646, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35278);