#105835

order	512
nilpotency class	3
isoclinism	isoclinic class of #8516
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

794 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 64

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

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

Polycyclic presentation:

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

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, 13031);