#501590

order	512
nilpotency class	4
isoclinism	isoclinic class of #501524
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=ehi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,e^c=ei,f^c=fh,h^c=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, 408786);