#102213

order	512
nilpotency class	3
isoclinism	isoclinic class of #102147
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#390
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

406 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9409);