#109825

order	512
nilpotency class	3
isoclinism	isoclinic class of #9415
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	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

431 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,10)(4,13)(5,8)(6,9)(7,11)(12,16)(14,15)(17,18,21,27,23,28,22,26)(19,29,31,24,33,39,32,40)(20,34,36,41,37,42,35,25)(30,44,46,43,47,38,45,48);(1,3,5,11,6,12,4,14)(2,7,8,16,9,15,13,10)(17,19,22,32,23,33,21,31)(18,24,27,39,28,40,26,29)(20,30,35,45,37,47,36,46)(25,38,34,48,41,44,42,43);(1,2)(3,7)(4,13)(5,8)(6,9)(10,14)(11,16)(12,15)(17,20,21,36,23,37,22,35)(18,25,27,34,28,41,26,42)(19,30,31,46,33,47,32,45)(24,38,39,48,40,44,29,43)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17021);