#419796

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1592 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326992);