#314647

order	512
nilpotency class	3
isoclinism	isoclinic class of #252247
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2482
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^b=fh,e^c=eh,f^c=fhi,g^c=gh,f^d=fh

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, 221843);