#148743

order	512
nilpotency class	4
isoclinism	isoclinic class of #134415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55939);