#420399

order	512
nilpotency class	3
isoclinism	isoclinic class of #419895
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1313 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=e,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,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, 327595);