#121808

order	512
nilpotency class	3
isoclinism	isoclinic class of #120186
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1445
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

682 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=gh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29004);