#101680

order	512
nilpotency class	3
isoclinism	isoclinic class of #101679
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

457 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8876);