#106198

order	512
nilpotency class	3
isoclinism	isoclinic class of #106194
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

736 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)(3,10,12,7)(4,8)(5,9)(6,11,14,16)(13,15)(17,18,21,27,23,28,22,26)(19,29,32,40,33,39,31,24)(20,34,35,42,36,41,37,25)(30,44,46,43,47,38,45,48);(1,3)(2,6,9,14)(4,11)(5,12)(7,8,10,15)(13,16)(17,19,22,31,23,33,21,32)(18,24,26,39,28,40,27,29)(20,30,35,46,36,47,37,45)(25,38,34,48,42,44,41,43);(1,3)(2,7)(4,11)(5,12)(6,8)(9,10)(13,16)(14,15)(17,20)(18,25)(19,30)(21,37)(22,35)(23,36)(24,38)(26,34)(27,41)(28,42)(29,43)(31,46)(32,45)(33,47)(39,48)(40,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13394);