#98034

order	512
nilpotency class	3
isoclinism	isoclinic class of #97894
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=i,d^2=h,e^2=g,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5230);