#121690

order	512
nilpotency class	3
isoclinism	isoclinic class of #102079
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,8)(4,6)(5,7)(9,10,11,14,12,13,15,16)(17,18,20,23,21,19,24,22);(1,3,5,4)(2,6,7,8)(10,13)(11,15)(17,19)(18,21)(20,23)(22,24);(1,4,5,3)(2,6,7,8)(10,13)(11,15)(18,22)(19,23)(20,24)

Polycyclic presentation:

2^9|a^2=g,b^2=dei,c^2=dei,d^2=i,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=ghi,d^c=di,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28886);