#94480

order	512
nilpotency class	5
isoclinism	isoclinic class of #94479
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7323
commutator	#117	C₄²
Frattini, cosocle	#2960	C₂²
prime radical	C₂³	#344
LCS	G ▷ #117 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2960 ◁ G
LECS	G ▷ #2960 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

145 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 145 conjugacy classes of subgroups — too many to draw. Only its 40 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=egh,h^a=hi,c^b=cg,d^b=dfh,f^b=fi,g^b=gh,d^c=di,e^c=ehi,g^c=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1676);