#113632

order	512
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#60951	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

815 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dh,e^c=ehi,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, 20828);