#94558

order	512
nilpotency class	6
isoclinism	isoclinic class of #94554
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7349
commutator	C₂⁵	C₂×C₈
Frattini, cosocle	#1783	C₂²
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂⁴ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1783 ◁ G
LECS	G ▷ #1783 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

214 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 24 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=g,d^2=f,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=de,e^b=ei,f^b=fh,d^c=dg,f^c=fi,e^d=eh,g^d=gi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1754);