#460668

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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=e,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, 367864);