#6370833

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4542 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)(5,6,10,15,11,16,9,14)(7,17,21,27,19,28,20,12)(8,22,23,13,25,29,24,30)(18,32,34,31,33,26,35,36);(5,7,9,20,11,19,10,21)(6,12,15,17,16,27,14,28)(8,18,23,34,25,33,24,35)(13,26,22,31,30,32,29,36);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,29)(15,22)(16,30)(17,31)(19,33)(20,34)(21,35)(27,32)(28,36);(7,19)(12,27)(17,28)(18,33)(20,21)(26,32)(31,36)(34,35);(5,9,11,10)(6,14,16,15)(7,20,19,21)(8,23,25,24)(12,28,27,17)(13,29,30,22)(18,34,33,35)(26,36,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317451, #72317468, #72317503
maximal normal subgroup of	#72317451, #72317468, #72317503

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278029);