#503501

order	512
nilpotency class	4
isoclinism	isoclinic class of #503499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25900
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#62328	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

799 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 799 conjugacy classes of subgroups — too many to draw. Only its 130 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)(3,14)(4,18,22,27)(5,8)(6,12)(7,13)(9,21,28,32)(10,17)(11,23)(15,25,30,19)(16,24)(20,26,31,29);(1,3,6,5,7,17,23,24)(2,8,11,14,13,16,12,10)(4,19,21,26,22,25,32,29)(9,15,27,20,28,30,18,31);(1,4,7,22)(2,9)(3,15,17,30)(5,20,24,31)(6,21,23,32)(8,25)(10,26)(11,27)(12,18)(13,28)(14,29)(16,19);(1,5,23,3,7,24,6,17)(2,10,12,16,13,14,11,8)(4,20,32,15,22,31,21,30)(9,26,18,19,28,29,27,25)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410697);