#147882

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=eh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55078);