#317924

order	512
nilpotency class	3
isoclinism	isoclinic class of #317914
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

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

normal subgroups only

This group has 621 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)(3,10)(4,6)(5,9)(7,12)(8,11)(13,14)(15,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,30)(27,29)(31,32);(2,6,9,8)(7,14,15,10)(17,19,21,26)(18,22,25,30)(20,27,28,32)(23,29,24,31);(1,2)(3,7)(4,8)(5,9)(6,11)(10,12)(13,15)(14,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,3,11,16,5,13,4,12)(2,7,6,14,9,15,8,10)(19,26)(22,30)(27,32)(29,31)

Polycyclic presentation:

2^9|b^2=eghi,d^2=eh,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=di,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, 225120);