#6372972

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60450	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2660 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)(5,6,11,16,12,17,10,28)(7,18,21,31,22,36,20,13)(8,14,25,33,23,32,24,30)(9,15,27,34,26,29,19,35);(5,7,10,20,12,22,11,21)(6,13,16,18,17,31,28,36)(8,19,24,26,23,27,25,9)(14,29,33,35,32,15,30,34);(2,4)(5,8)(6,14)(7,19)(9,21)(10,24)(11,25)(12,23)(13,29)(15,31)(16,33)(17,32)(18,35)(20,26)(22,27)(28,30)(34,36);(5,9,11,27,12,26,10,19)(6,15,16,34,17,29,28,35)(7,8,21,25,22,23,20,24)(13,30,18,14,31,33,36,32);(8,23)(9,26)(14,32)(15,29)(19,27)(24,25)(30,33)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280168);