#472030

order	512
nilpotency class	3
isoclinism	isoclinic class of #469776
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

1318 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)(7,16,19,34)(8,13)(9,23,25,12)(10,14)(11,15)(17,33)(18,35,20,29)(21,30)(22,31)(24,27,36,28)(26,32);(1,2,3,4)(5,7,17,25,11,19,10,9)(6,12,14,16,15,23,33,34)(8,20,21,24,22,18,26,36)(13,28,32,35,31,27,30,29);(2,4)(5,8,11,22)(6,13,15,31)(7,18,19,20)(9,24,25,36)(10,26,17,21)(12,27,23,28)(14,32,33,30)(16,29,34,35);(5,9,10,19,11,25,17,7)(6,12,14,16,15,23,33,34)(8,20,21,24,22,18,26,36)(13,29,30,27,31,35,32,28)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379226);