#391250

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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

921 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:

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

Polycyclic presentation:

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

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, 298446);