#459144

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

930 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 366340);