#248446

order	512
nilpotency class	3
isoclinism	isoclinic class of #248359
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61707	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fhi,d^c=di,f^c=fi,e^d=eh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155642);