#106453

order	512
nilpotency class	3
isoclinism	isoclinic class of #106434
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

921 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=dh,f^b=fi,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, 13649);