#427838

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

1231 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=ef,f^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=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335034);