#498371

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×#472	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

616 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)(5,6,12,17,33,16,34,23,13,19,10,7,11,18,28,22)(8,9,15,31,25,14,27,35,26,32,24,29,21,30,36,20);(1,3,4,2)(5,7,13,17)(6,12,19,10)(8,20,26,35)(9,27,32,36)(11,22,33,23)(14,15,30,24)(16,28,18,34)(21,31,25,29);(5,8)(6,14)(7,20)(9,18)(10,24)(11,21)(12,15)(13,26)(16,32)(17,35)(19,30)(22,31)(23,29)(25,33)(27,34)(28,36);(5,9,28,20,11,30,10,29,13,32,34,35,33,14,12,31)(6,15,22,8,18,36,7,21,19,24,23,26,16,27,17,25)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=ehi,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, 405567);