#6382008

order	512
nilpotency class	3
isoclinism	isoclinic class of #61106
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,f^b=fi,h^b=hi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289204);