#497462

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#3415	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

981 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)(5,6,10,15,12,8,11,23)(7,16,20,33,21,34,29,18)(9,24,26,22,28,36,27,14)(13,25,30,32,17,35,31,19);(5,7)(6,13)(8,17)(9,25)(10,20)(11,29)(12,21)(14,18)(15,30)(16,24)(19,27)(22,33)(23,31)(26,32)(28,35)(34,36);(2,4)(5,8)(6,12)(7,17)(9,22)(10,15)(11,23)(13,21)(14,28)(16,32)(18,35)(19,34)(20,30)(24,26)(25,33)(27,36)(29,31);(5,6)(7,18)(8,12)(9,24)(10,15)(11,23)(13,25)(14,27)(16,20)(17,35)(19,31)(21,33)(22,26)(28,36)(29,34)(30,32)

Polycyclic presentation:

2^9|a^2=g,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,f^c=fi,g^c=gi,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, 404658);