#197551

order	512
nilpotency class	3
isoclinism	isoclinic class of #197339
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#34633	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^2=i,e^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,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, 104747);