#410408

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

809 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)(7,17,19,28,20,12,25,27)(8,21,23,13,22,30,24,31)(18,32,34,29,33,35,36,26);(2,4)(5,7,10,19,11,20,9,25)(6,12,14,27,15,17,16,28)(8,18,24,34,22,33,23,36)(13,29,21,35,31,26,30,32);(1,2,3,4)(5,8)(6,13,15,31)(7,18)(9,23)(10,24)(11,22)(12,26,17,29)(14,21,16,30)(19,34)(20,33)(25,36)(27,32,28,35);(6,14,15,16)(8,22)(12,27,17,28)(13,30,31,21)(18,33)(23,24)(26,35,29,32)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317604);