#120698

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=di,d^2=h,e^2=i,f^2=hi,g^2=hi,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fh,g^b=gh,e^c=ei,f^c=fi,g^c=gi

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, 27894);