#211613

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

723 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=di,d^c=dh,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, 118809);