#106763

order	512
nilpotency class	3
isoclinism	isoclinic class of #106738
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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213796, #72213874, #72213906, …
maximal normal subgroup of	#72213796, #72213874, #72213906, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13959);