#498167

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#576	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

746 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=deh,e^b=eh,h^b=hi,f^c=fi,e^d=eh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405363);