#391651

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂×#392	C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1454 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 1454 conjugacy classes of subgroups — too many to draw. Only its 254 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=fg,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,d^c=dei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298847);