#460669

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

892 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=e,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,e^d=ei,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, 367865);