#107560

order	512
nilpotency class	3
isoclinism	isoclinic class of #107542
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

742 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=di,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, 14756);