#106347

order	512
nilpotency class	3
isoclinism	isoclinic class of #106326
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	C₂²×#41	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

805 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^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	#72316632, #72316649
maximal normal subgroup of	#72316632, #72316649

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13543);