#491658

order	512
nilpotency class	4
isoclinism	isoclinic class of #491653
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

574 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=ci,d^c=dg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398854);