#391856

order	512
nilpotency class	3
isoclinism	isoclinic class of #213555
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1416 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 242 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299052);