#112354

order	512
nilpotency class	3
isoclinism	isoclinic class of #97642
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19550);