#119229

order	512
nilpotency class	3
isoclinism	isoclinic class of #99947
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

736 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:

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

Polycyclic presentation:

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

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, 26425);