#146668

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×#525	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

1098 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)(7,15,16,30)(8,19,21,13)(9,22,23,12)(10,14)(11,17)(18,28)(20,27)(24,36,25,33)(26,31)(29,32)(34,35);(1,2,3,4)(5,7)(6,11,14,26)(8,17)(9,18)(10,16)(12,25,22,24)(13,15,19,30)(20,33)(21,31)(23,32)(27,29,34,28)(35,36);(6,12)(7,16)(11,24)(13,27)(14,22)(15,28)(17,31)(18,32)(19,34)(25,26)(29,30)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280868, #72282288, #72282797
maximal normal subgroup of	#72280868, #72282288, #72282797

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53864);