#457208

order	512
nilpotency class	3
isoclinism	isoclinic class of #456417
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1307 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:

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

Polycyclic presentation:

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

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, 364404);