#106195

order	512
nilpotency class	3
isoclinism	isoclinic class of #106194
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

907 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316682, #72316926, #72316929
maximal normal subgroup of	#72316682, #72316926, #72316929
maximal quotient of	#72279289

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13391);