#430451

order	512
nilpotency class	3
isoclinism	isoclinic class of #422233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1225 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,d^2=efi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,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, 337647);