#107653

order	512
nilpotency class	3
isoclinism	isoclinic class of #107630
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

710 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14849);