#109889

order	512
nilpotency class	3
isoclinism	isoclinic class of #9431
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

491 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17085);