#100083

order	512
nilpotency class	3
isoclinism	isoclinic class of #100075
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

666 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cgi,d^b=dh,e^b=eh,f^b=fh,d^c=dh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7279);