#112656

order	512
nilpotency class	3
isoclinism	isoclinic class of #112637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

827 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19852);