#121720

order	512
nilpotency class	3
isoclinism	isoclinic class of #120247
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#391
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

484 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=i,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,g^b=ghi,d^c=di,e^c=ei,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, 28916);