#501019

order	512
nilpotency class	4
isoclinism	isoclinic class of #501014
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

998 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi

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, 408215);