#263449

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	#62373	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 817 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)(19,23)(20,28)(21,30)(22,26)(24,29)(25,31)(27,32);(1,2,5,15,6,9,4,8)(3,7,13,16,14,11,12,10)(17,19,22,20)(18,23,26,28)(21,29,31,27)(24,25,32,30);(1,2)(3,11)(4,8)(5,15)(6,9)(7,14)(10,13)(12,16)(17,20,22,19)(18,24,26,32)(21,27,31,29)(23,30,28,25);(1,3,5,13,6,14,4,12)(2,7,15,16,9,11,8,10)

Polycyclic presentation:

2^9|b^2=ehi,c^2=i,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,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, 170645);