#494302

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,h^a=hi,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fhi,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, 401498);