#99793

order	512
nilpotency class	3
isoclinism	isoclinic class of #9159
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6989);