#146733

order	512
nilpotency class	4
isoclinism	isoclinic class of #12741
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

984 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)(7,17,19,27)(8,11)(9,25)(10,16)(12,18,29,35)(13,21)(14,32)(15,23)(20,31,26,33)(22,30,36,34)(24,28);(1,2,3,4)(6,11)(8,21)(9,23)(12,27)(13,15)(14,25)(16,28)(17,29)(18,22)(20,26)(24,32)(30,33)(31,34)(35,36);(5,7)(6,12)(8,22)(9,26)(10,19)(11,27)(13,31)(14,33)(15,34)(16,29)(17,28)(18,21)(20,23)(24,36)(25,30)(32,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dh,e^b=eh,h^b=hi,d^c=dh,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, 53929);