#498166

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×#2193	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

742 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)(5,6,11,18)(7,20,24,28)(8,15,27,34)(9,21,30,35)(10,19,12,17)(13,33,26,22)(14,32,25,23)(16,29,36,31);(1,3,4,2)(5,7,11,24)(6,13,18,26)(8,19,27,17)(9,15,30,34)(10,23,12,32)(14,16,25,36)(20,35,28,21)(22,29,33,31);(5,7,11,24)(6,14,18,25)(8,21,27,35)(9,22,30,33)(10,32,12,23)(13,16,26,36)(15,29,34,31)(17,28,19,20);(5,8)(6,15)(7,21)(9,28)(10,26)(11,27)(12,13)(14,29)(16,23)(17,33)(18,34)(19,22)(20,30)(24,35)(25,31)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^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, 405362);