#146710

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂²	#1782
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1090 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)(7,16,19,28)(8,13,21,30)(9,14,24,12)(10,15)(11,17)(18,33)(20,26)(22,29)(23,32)(25,35,31,34)(27,36);(1,2,3,4)(5,7)(6,11,15,26)(8,20)(9,23)(10,19)(12,25,14,31)(13,28,30,16)(17,21)(18,24)(22,34)(27,32,29,33)(35,36);(6,12)(8,21)(9,24)(11,25)(13,29)(14,15)(16,32)(17,20)(18,23)(26,31)(27,30)(28,33)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fhi,f^d=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279249, #72279536, #72281167, …
maximal normal subgroup of	#72279249, #72279536, #72281167, …
maximal quotient of	#72279296

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53906);