#146903

order	512
nilpotency class	4
isoclinism	isoclinic class of #12826
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=dh,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54099);