#146770

order	512
nilpotency class	4
isoclinism	isoclinic class of #12750
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

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250063, #72250068, #72250079, …
maximal normal subgroup of	#72250063, #72250068, #72250079, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53966);