#121251

order	512
nilpotency class	3
isoclinism	isoclinic class of #121250
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=dh,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=ghi,e^c=ehi,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, 28447);