#147217

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,3)(9,10)(11,22)(12,26)(13,20)(14,19)(15,21)(16,23)(17,24)(18,29)(25,37)(27,38)(28,32)(30,36)(31,35)(33,40)(34,39);(5,6,7,8)(9,11)(10,16)(12,27)(13,24)(14,31)(15,25)(17,32)(18,35)(19,33)(20,38)(21,34)(22,26)(23,29)(28,39)(30,40)(36,37);(1,3,4,2)(9,11)(10,17)(12,23)(13,31)(14,24)(15,25)(16,32)(18,34)(19,37)(20,22)(21,35)(26,38)(27,29)(28,40)(30,39)(33,36)

Polycyclic presentation:

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

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, 54413);