#495063

order	512
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#459
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

688 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)(7,18)(8,16)(9,15)(10,25)(11,27)(12,17)(13,19)(14,23)(20,34)(21,32)(22,28)(24,33)(26,29)(30,35)(31,36);(1,3,4,2)(5,7)(6,13)(8,19)(9,20)(10,26)(11,21)(12,22)(14,28)(15,29)(16,32)(17,31)(18,25)(23,35)(24,36)(27,30)(33,34);(5,8,11,23,12,24,9,10)(6,14,15,16,17,25,27,33)(7,19,21,35,22,36,20,26)(13,28,29,32,31,18,30,34);(5,9,12,11)(6,15,17,27)(7,20,22,21)(8,10,24,23)(13,29,31,30)(14,16,25,33)(18,34,28,32)(19,26,36,35)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402259);