#406115

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1001 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313311);