#494314

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

1066 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)(5,6)(7,18)(8,12)(9,27)(10,28)(11,14)(13,20)(15,23)(16,26)(17,25)(19,33)(21,32)(22,34)(24,31)(29,30)(35,36);(1,3,4,2)(6,12)(8,23)(9,26)(10,25)(13,30)(14,16)(15,17)(18,33)(19,20)(21,29)(24,36)(27,28)(31,32)(34,35);(5,7)(6,13)(8,24)(9,19)(10,29)(11,22)(12,30)(14,31)(15,34)(16,32)(17,35)(18,27)(20,26)(21,25)(23,36)(28,33);(6,14)(12,16)(13,31)(15,27)(17,28)(18,34)(30,32)(33,35)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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=fhi,h^b=hi,e^c=ehi,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, 401510);