#456885

order	512
nilpotency class	3
isoclinism	isoclinic class of #456884
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1130 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:

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

Polycyclic presentation:

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

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, 364081);