#500669

order	512
nilpotency class	4
isoclinism	isoclinic class of #499919
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	#60643	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1172 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)(7,19)(8,13)(9,18)(10,16)(11,26)(12,15)(14,23)(17,28)(20,30)(21,33)(22,34)(24,35)(25,29)(27,31)(32,36);(1,2,4,3)(5,7,10,8)(6,13,16,19)(9,22,12,20)(11,23,25,21)(14,26,33,29)(15,34,18,30)(17,32,35,31)(24,36,28,27);(5,8,10,7)(6,14,16,33)(9,22,12,20)(11,27,25,36)(13,29,19,26)(15,32,18,31)(17,30,35,34)(21,28,23,24);(1,3,4,2)(5,9,10,12)(6,15,16,18)(7,20,8,22)(11,24,25,28)(13,30,19,34)(14,32,33,31)(17,29,35,26)(21,36,23,27)

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,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, 407865);