#106895

order	512
nilpotency class	3
isoclinism	isoclinic class of #106878
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

832 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)(3,11,4,7)(5,10)(6,9)(8,13,15,12)(14,16)(17,18,20,23,21,24,28,31)(19,25,27,32,26,22,29,30);(1,3,6,4)(2,7,9,11)(5,12,14,13)(8,10,15,16)(17,19)(18,22,24,25)(20,27)(21,26)(23,30,31,32)(28,29);(1,4,6,3)(2,8,9,15)(5,12,14,13)(7,10,11,16)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14091);