#106312

order	512
nilpotency class	3
isoclinism	isoclinic class of #1909
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

993 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gh,e^b=eh,d^c=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13508);