#93783

order	512
nilpotency class	5
isoclinism	isoclinic class of #93782
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂×C₄⋊C₈	C₂×C₄
Frattini, cosocle	#1984	C₂²
prime radical	#3245	C₄
LCS	G ▷ C₂×C₄⋊C₈ ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1984 ◁ G
LECS	G ▷ #1984 ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁

205 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 32 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,c^2=fhi,d^2=i,e^2=h,f^2=hi,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,d^b=defh,e^b=eh,f^b=fhi,g^b=gi,d^c=dgh,e^c=ei,e^d=ei

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, 979);