#106788

order	512
nilpotency class	3
isoclinism	isoclinic class of #106766
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	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1049 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)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16)(17,18,20,23,21,24,28,31)(19,25,27,32,26,22,29,30);(1,3)(2,6,8,14)(4,9)(5,10)(7,13,11,16)(12,15)(17,19)(18,22,24,25)(20,27)(21,26)(23,30,31,32)(28,29);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13984);