#115391

order	512
nilpotency class	3
isoclinism	isoclinic class of #115390
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,d^c=di,e^c=ei,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257776, #72257781
maximal normal subgroup of	#72257776, #72257781
maximal quotient of	#72257797

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22587);