#423174

order	512
nilpotency class	3
isoclinism	isoclinic class of #419839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1333 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 330370);