#102021

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#390
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

408 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9217);