#509616

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25441	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

639 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=ehi,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=eh,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416812);