#107620

order	512
nilpotency class	3
isoclinism	isoclinic class of #8491
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,d^b=di,f^b=fhi,e^c=eh,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14816);