#474269

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2572
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

846 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=efgh,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=de,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381465);