#391209

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

1077 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298405);