#498317

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×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

618 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)(5,6,11,17,12,18,9,16)(7,19,22,13,23,29,21,30)(8,15,25,34,26,35,10,27)(14,31,28,36,33,24,32,20);(1,3,4,2)(5,7)(6,13)(8,24)(9,22)(10,20)(11,21)(12,23)(14,15)(16,29)(17,19)(18,30)(25,36)(26,31)(27,28)(32,34)(33,35);(5,7)(6,14)(8,20)(9,21)(10,24)(11,22)(12,23)(13,15)(16,32)(17,28)(18,33)(19,27)(25,31)(26,36)(29,34)(30,35);(5,8,11,25,12,26,9,10)(6,15,17,34,18,35,16,27)(7,20,22,31,23,36,21,24)(13,28,29,33,30,32,19,14)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,e^b=ei,g^b=gi,f^c=fh,h^c=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, 405513);