#121786

order	512
nilpotency class	3
isoclinism	isoclinic class of #120577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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

492 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=ei,g^b=gi,d^c=dhi,e^c=ehi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28982);