#318593

order	512
nilpotency class	3
isoclinism	isoclinic class of #318583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 615 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)(3,8)(4,13)(5,15)(6,10)(7,12)(9,14)(11,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,32)(28,30)(29,31);(1,3,5,12,6,11,14,4)(2,7,9,8,10,13,15,16)(17,19,20,27,22,28,21,29)(18,23,24,32,26,30,25,31);(3,11)(4,12)(7,13)(8,16)(18,24)(19,27)(20,21)(23,30)(25,26)(28,29);(1,4,14,11,6,12,5,3)(2,8,15,7,10,16,9,13)(19,28)(23,30)(27,29)(31,32)

Polycyclic presentation:

2^9|b^2=eghi,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225789);