#465623

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1085 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,h^b=hi,d^c=de,f^c=fi,e^d=ei,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, 372819);