#391218

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
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

1122 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:

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

Polycyclic presentation:

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

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, 298414);