#110099

order	512
nilpotency class	3
isoclinism	isoclinic class of #110086
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

717 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh

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, 17295);