#106187

order	512
nilpotency class	3
isoclinism	isoclinic class of #106181
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	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18)(19,24,29,39)(20,31,32,42)(21,27)(22,36)(23,26)(25,33,41,35)(28,43)(30,38,34,40)(37,45)(44,48)(46,47);(17,19,22,30,23,29,21,34)(18,24,27,40,26,39,36,38)(20,28,33,45,32,46,35,44)(25,37,42,43,41,48,31,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,20)(18,25)(19,28)(21,35)(22,33)(23,32)(24,37)(26,41)(27,42)(29,46)(30,45)(31,36)(34,44)(38,47)(39,48)(40,43)

Polycyclic presentation:

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

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, 13383);