#118895

order	512
nilpotency class	3
isoclinism	isoclinic class of #10631
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

686 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)(13,14,18,24,19,22,17,23)(15,25,28,36,29,20,27,37)(16,30,32,39,34,40,33,21)(26,42,43,35,44,38,31,41);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15,19,29)(14,20,22,25)(16,31,34,43)(17,28,18,27)(21,38,39,42)(23,37,24,36)(26,32,44,33)(30,35,40,41);(2,4)(5,7,8,6,10,11,9,12)(13,16)(14,21,22,39)(15,26)(17,32)(18,33)(19,34)(20,35,25,41)(23,30,24,40)(27,43)(28,31)(29,44)(36,38,37,42)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26091);