#120749

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
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	#121
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

682 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fi,g^b=gh,f^c=fhi

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, 27945);