#115201

order	512
nilpotency class	3
isoclinism	isoclinic class of #9938
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#2404	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1019 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,g^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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257776, #72257777, #72257780
maximal normal subgroup of	#72257776, #72257777, #72257780
maximal quotient of	#72257805

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22397);