#148434

order	512
nilpotency class	4
isoclinism	isoclinic class of #13334
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

692 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,d^c=dhi,f^c=fhi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321678, #72321712, #72321727, …
maximal normal subgroup of	#72321678, #72321712, #72321727, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55630);