#99964

order	512
nilpotency class	3
isoclinism	isoclinic class of #9303
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

472 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:

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

Polycyclic presentation:

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

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, 7160);