#107509

order	512
nilpotency class	3
isoclinism	isoclinic class of #107494
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:

(2,7,9,12)(8,15,16,14)(17,18,21,27,23,28,22,26)(19,29,31,24,33,39,32,40)(20,34,36,41,37,42,35,25)(30,44,46,43,47,38,45,48);(1,2,5,7,6,9,4,12)(3,8,10,14,11,16,13,15)(17,19,22,32,23,33,21,31)(18,24,27,39,28,40,26,29)(20,30,35,45,37,47,36,46)(25,38,34,48,41,44,42,43);(1,3)(2,8)(4,13)(5,10)(6,11)(7,14)(9,16)(12,15)(17,20)(18,25)(19,30)(21,36)(22,35)(23,37)(24,38)(26,42)(27,34)(28,41)(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=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=di,e^b=ehi,f^b=fi,d^c=dhi,e^c=eh

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, 14705);