#94708

order	512
nilpotency class	6
isoclinism	isoclinic class of #94676
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1412
commutator	#376	C₂×C₄
Frattini, cosocle	#1963	C₂²
socle	C₂	#7428
prime radical	#532	D₄
LCS	G ▷ #376 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1963 ◁ G
LECS	G ▷ #1963 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 143 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgi,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=defghi,g^b=gi,d^c=dghi,e^c=ei,e^d=ehi,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1904);