#105967

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#380
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

854 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:

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

Polycyclic presentation:

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

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, 13163);