#94570

order	512
nilpotency class	6
isoclinism	isoclinic class of #7529
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1408
commutator	C₄×C₈	C₂×C₈
Frattini, cosocle	#1848	C₂²
socle	C₂	#7354
prime radical	#3147	C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1848 ◁ G
LECS	G ▷ #1848 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

158 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 26 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,c^2=gh,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cghi,d^b=deg,e^b=ehi,g^b=gi,d^c=dgh,e^d=ehi,g^d=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, 1766);