#147668

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×#3058	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1120 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)(9,10,13,18,14,20,31,19)(11,21,23,35,25,37,24,16)(12,26,29,17,30,15,27,33)(22,36,38,40,28,34,39,32);(5,6)(7,8)(10,15)(12,27)(13,31)(16,32)(17,18)(19,33)(20,26)(21,34)(22,38)(23,24)(28,39)(29,30)(35,40)(36,37);(2,4)(6,8)(9,11)(10,16)(12,28)(13,24)(14,25)(15,32)(17,36)(18,37)(19,21)(20,35)(22,30)(23,31)(26,40)(27,39)(29,38)(33,34)

Polycyclic presentation:

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

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, 54864);