#110215

order	512
nilpotency class	3
isoclinism	isoclinic class of #110189
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17411);