#119108

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#397
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

726 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26304);