#146620

order	512
nilpotency class	4
isoclinism	isoclinic class of #12767
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×D₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1031 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,e^b=ehi,f^b=fh,d^c=dh,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53816);