#512657

order	512
nilpotency class	5
isoclinism	isoclinic class of #512655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	#3498	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3498 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

830 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=h,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=eh,f^b=fghi,g^b=gi,d^c=di,e^c=ei,f^e=fhi,g^f=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, 419853);