#105403

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bd,d^a=dh,f^a=fhi,c^b=ch,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12599);