#112622

order	512
nilpotency class	3
isoclinism	isoclinic class of #10088
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

833 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)(3,10,12,7)(4,8)(5,9)(6,11,16,13)(14,15)(17,18)(19,27,30,23)(20,31)(21,34)(22,25)(24,33)(26,37)(28,44,45,48)(29,39,35,41)(32,42)(36,40)(38,46,43,47);(1,3)(2,6)(4,13)(5,12)(7,8)(9,16)(10,15)(11,14)(17,19,22,30)(18,23,25,27)(20,28,32,45)(21,35,37,29)(24,38,40,43)(26,41,34,39)(31,44,42,48)(33,47,36,46);(1,3)(2,7)(4,13)(5,12)(6,15)(8,16)(9,10)(11,14)(17,20)(18,24)(19,28)(21,36)(22,32)(23,38)(25,40)(26,42)(27,43)(29,47)(30,45)(31,34)(33,37)(35,46)(39,44)(41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,e^c=eh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19818);