#498369

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×#2221	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

629 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)(5,6,10,16,11,32,30,36,13,18,31,25,12,17,26,35)(7,19,20,33,24,9,15,27,8,22,23,34,21,28,29,14);(1,3,4,2)(5,7)(6,14)(8,13)(9,25)(10,29)(11,21)(12,24)(15,31)(16,28)(17,33)(18,27)(19,35)(20,26)(22,36)(23,30)(32,34);(5,8)(6,9)(7,13)(10,23)(11,21)(12,24)(14,25)(15,26)(16,27)(17,19)(18,28)(20,31)(22,32)(29,30)(33,35)(34,36);(5,9,26,33,12,19,31,14,13,28,30,34,11,22,10,27)(6,15,35,24,17,20,25,7,18,29,36,21,32,23,16,8)

Polycyclic presentation:

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

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, 405565);