#263437

order	512
nilpotency class	3
isoclinism	isoclinic class of #263431
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10300	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

798 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 798 conjugacy classes of subgroups — too many to draw. Only its 144 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)(4,15)(5,8)(6,9)(7,12)(11,13)(14,16)(17,18)(19,22)(20,23)(21,25)(24,29)(26,30)(27,32)(28,31);(1,2,5,15,6,9,4,8)(3,7,13,16,14,11,12,10)(17,19,21,27)(18,22,25,32)(20,28,29,26)(23,31,24,30);(1,2)(3,11)(4,8)(5,15)(6,9)(7,14)(10,13)(12,16)(18,23)(20,29)(22,30)(24,25)(26,28)(31,32);(1,3,4,12,6,14,5,13)(2,7,8,10,9,11,15,16)

Polycyclic presentation:

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

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, 170633);