#391514

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1062 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 266 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=g,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=defi,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, 298710);