#107444

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

741 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14640);