#419898

order	512
nilpotency class	3
isoclinism	isoclinic class of #419895
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1551 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)(7,18)(8,14)(9,25,28,34)(10,19)(11,16)(12,17)(13,21)(15,29,33,27)(20,23,35,30)(22,24)(26,32,31,36);(1,2)(3,4)(5,7,12,22)(6,13,17,8)(9,26,28,31)(10,14,11,21)(15,32,33,36)(16,24,19,18)(20,29,35,27)(23,25,30,34);(2,4)(5,8,12,13)(6,14,17,21)(7,19,22,16)(9,23,28,30)(10,24,11,18)(15,31,33,26)(20,34,35,25)(27,36,29,32);(5,9,10,27,12,28,11,29)(6,15,16,25,17,33,19,34)(7,20,21,26,22,35,14,31)(8,23,24,36,13,30,18,32)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,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, 327094);