#376193

order	512
nilpotency class	3
isoclinism	isoclinic class of #219491
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ehi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi

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, 283389);