#471445

order	512
nilpotency class	3
isoclinism	isoclinic class of #456658
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1127 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:

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

Polycyclic presentation:

2^9|b^2=egh,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi,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, 378641);