#503500

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	#3549	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 767 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)(3,10)(4,19,23,27)(5,17)(6,12)(7,13)(8,16)(9,22,28,29)(11,14)(15,24,31,20)(18,25)(21,26,30,32);(1,3,14,5,7,18,6,16)(2,8,12,25,13,17,11,10)(4,20,29,26,23,24,22,32)(9,15,19,21,28,31,27,30);(1,4,7,23)(2,9)(3,15,18,31)(5,21,16,30)(6,22,14,29)(8,24)(10,26)(11,27)(12,19)(13,28)(17,20)(25,32);(1,5,6,3,7,16,14,18)(2,10,11,17,13,25,12,8)(4,21,22,15,23,30,29,31)(9,26,27,20,28,32,19,24)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=ef,d^2=e,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, 410696);