#6369867

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂×#585	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,f^2=i,g^2=i,b^a=bf,f^a=fi,h^a=hi,e^d=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226779, #72314246, #72317429
maximal normal subgroup of	#72226779, #72314246, #72317429

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277063);