#6371885

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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2592 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)(5,6,11,16,12,17,10,26)(7,13,19,28,20,14,8,21)(9,15,25,31,23,30,24,35)(18,27,33,36,32,29,22,34);(2,4)(5,7,11,19,12,20,10,8)(6,13,16,28,17,14,26,21)(9,18,25,33,23,32,24,22)(15,27,31,36,30,29,35,34);(5,8,10,20,12,19,11,7)(6,14,16,21,17,13,26,28)(9,22,24,32,23,33,25,18)(15,29,31,34,30,27,35,36);(5,9)(6,15)(7,18)(8,22)(10,24)(11,25)(12,23)(13,27)(14,29)(16,31)(17,30)(19,33)(20,32)(21,34)(26,35)(28,36);(9,23)(15,30)(18,32)(22,33)(24,25)(27,29)(31,35)(34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,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, 6279081);