#209705

order	512
nilpotency class	3
isoclinism	isoclinic class of #18682
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#550	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1062 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)(5,6,9,13)(7,11,14,18)(8,10,16,12)(15,17,20,19)(21,22,25,28)(23,29,30,36)(24,32,33,27)(26,34,35,31);(6,10)(8,16)(11,17)(12,13)(15,20)(18,19)(24,33)(27,32)(29,36)(31,34);(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20)(21,23,25,30)(22,26,28,35)(24,34,33,31)(27,36,32,29);(1,3,4,2)(7,14)(11,18)(15,20)(17,19)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116901);