#419850

order	512
nilpotency class	3
isoclinism	isoclinic class of #419450
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1582 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=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, 327046);