#108215

order	512
nilpotency class	3
isoclinism	isoclinic class of #9471
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

798 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,4,9,6,10,5,8)(3,11,13,7,12,16,14,15)(18,23)(20,21)(24,32)(25,26)(27,31)(28,30);(1,3,5,14,6,12,4,13)(2,7,9,16,10,15,8,11)(17,18,21,26,22,25,20,23)(19,27,28,31,29,24,30,32);(3,12)(7,15)(11,16)(13,14)(17,19)(18,24)(20,30)(21,28)(22,29)(23,31)(25,27)(26,32)

Polycyclic presentation:

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

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, 15411);