#475694

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

810 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)(7,8)(9,24,27,36)(10,29)(11,15)(12,16)(13,18)(14,28,32,26)(17,25,33,30)(19,22)(20,23)(21,34,35,31);(1,2,3,4)(5,7,10,18,12,20,11,19)(6,13,15,8,16,22,29,23)(9,17,26,31,27,33,28,34)(14,30,24,21,32,25,36,35);(2,4)(5,8,11,22,12,23,10,13)(6,7,15,19,16,20,29,18)(9,25,28,35,27,30,26,21)(14,31,24,17,32,34,36,33);(5,9,10,26,12,27,11,28)(6,14,15,24,16,32,29,36)(7,17,18,31,20,33,19,34)(8,21,22,25,23,35,13,30)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,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, 382890);