#106330

order	512
nilpotency class	3
isoclinism	isoclinic class of #106326
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

856 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)(3,10,12,16)(4,8)(5,14)(6,9)(7,11,15,13)(17,18)(19,25,26,23)(20,21)(22,24)(27,31,32,30)(28,29);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(17,19,22,27,21,26,29,32)(18,23,24,30,20,25,28,31);(17,20,21,18)(19,25,26,23)(22,28,29,24)(27,31,32,30)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302150, #72302153, #72302154
maximal normal subgroup of	#72302150, #72302153, #72302154

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13526);