#106172

order	512
nilpotency class	3
isoclinism	isoclinic class of #8359
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

(5,6)(7,10)(8,11)(9,12)(13,14,16,20,17,19,25,27)(15,21,22,28,23,18,24,26);(1,2,3,4)(6,10)(7,8)(11,12);(6,11)(7,8)(10,12)(13,15,17,23)(14,18,19,21)(16,24,25,22)(20,28,27,26)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13368);