#478498

order	512
nilpotency class	3
isoclinism	isoclinic class of #476707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

805 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)(5,6,11,16,10,13,9,15)(7,17,19,28,21,32,20,12)(8,22,26,14,25,31,24,29)(18,27,33,30,34,35,23,36);(5,7,9,19,10,21,11,20)(6,12,16,32,13,28,15,17)(8,23,26,34,25,33,24,18)(14,30,22,35,29,36,31,27);(6,13)(8,24)(9,11)(14,31)(17,32)(18,33)(19,20)(22,29)(23,34)(25,26)(27,30)(35,36);(2,4)(5,8,9,24,10,25,11,26)(6,14,15,22,13,29,16,31)(7,18,19,33,21,34,20,23)(12,27,17,30,28,35,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385694);