#317944

order	512
nilpotency class	3
isoclinism	isoclinic class of #317935
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3245	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

550 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=egi,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dhi,f^c=fi,g^c=gh

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, 225140);