#469245

order	512
nilpotency class	3
isoclinism	isoclinic class of #459488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

928 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)(5,6,11,13,10,16,9,15)(7,17,25,28,18,27,20,12)(8,21,24,14,23,29,22,30)(19,26,33,31,32,36,34,35);(5,7)(6,12,16,28)(8,19)(9,20)(10,18)(11,25)(13,17,15,27)(14,26,30,36)(21,35,29,31)(22,34)(23,32)(24,33);(1,2,4,3)(6,13)(7,18)(8,22)(9,11)(12,17)(14,30)(15,16)(19,33)(23,24)(26,36)(27,28)(32,34);(1,3,4,2)(5,8,9,22,10,23,11,24)(6,14,15,21,16,30,13,29)(7,19,20,34,18,32,25,33)(12,26,27,35,28,36,17,31)

Polycyclic presentation:

2^9|a^2=eg,b^2=gi,c^2=efg,d^2=f,e^2=i,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,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi

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, 376441);