#478600

order	512
nilpotency class	3
isoclinism	isoclinic class of #476707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

807 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)(5,6,10,15,11,14,9,25)(7,12,18,28,19,29,17,27)(8,20,24,31,22,36,23,13)(16,32,21,26,33,30,34,35);(5,7)(6,12)(8,21,22,34)(9,17)(10,18)(11,19)(13,30,31,32)(14,29)(15,28)(16,24,33,23)(20,35,36,26)(25,27);(5,8,9,23,11,22,10,24)(6,13,15,20,14,31,25,36)(7,16,18,21,19,33,17,34)(12,26,27,32,29,35,28,30);(2,4)(6,14)(7,17)(8,22)(9,10)(12,27)(16,21)(18,19)(20,36)(26,30)(28,29)(32,35)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385796);