#353666

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^b=gi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255607, #72255612, #72255634, …
maximal normal subgroup of	#72255607, #72255612, #72255634, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260862);