#456512

order	512
nilpotency class	3
isoclinism	isoclinic class of #456510
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1164 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,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, 363708);