#498372

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
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

614 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)(5,6,10,17,11,31,30,22,13,19,14,7,12,18,27,33)(8,23,16,34,32,29,36,20,24,9,25,28,21,15,26,35);(1,3,4,2)(5,7)(6,14)(8,20)(9,26)(10,19)(11,22)(12,33)(13,17)(15,25)(16,29)(18,27)(21,28)(23,36)(24,35)(30,31)(32,34);(5,8)(6,15)(7,20)(9,18)(10,16)(11,32)(12,21)(13,24)(14,25)(17,35)(19,29)(22,34)(23,31)(26,27)(28,33)(30,36);(5,9,27,20,12,29,14,34,13,23,30,35,11,15,10,28)(6,16,33,8,18,26,7,21,19,25,22,24,31,36,17,32)

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,d^a=di,f^a=fi,d^b=de,e^b=eh,g^b=gh,h^b=hi,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405568);