#500723

order	512
nilpotency class	4
isoclinism	isoclinic class of #499754
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61038	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1073 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,f^c=fh,g^c=gh,h^c=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, 407919);