#512888

order	512
nilpotency class	5
isoclinism	isoclinic class of #512884
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2196
commutator, Frattini, cosocle	#124	C₂⁴
prime radical	#61111	C₂
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1016 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 112 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420084);