#209618

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
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×#562	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

957 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)(5,6,10,14)(7,15,17,19)(8,12,9,13)(11,18,20,16)(21,22,26,24)(23,29,31,36)(25,33,35,28)(27,34,30,32);(5,7)(6,11)(8,18)(9,16)(10,17)(12,15)(13,19)(14,20)(21,23)(22,27)(24,30)(25,34)(26,31)(28,36)(29,33)(32,35);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(21,24,26,22)(23,30,31,27)(25,28,35,33)(29,34,36,32);(1,3,4,2)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,20,15)(23,31)(27,30)(29,36)(32,34)

Polycyclic presentation:

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

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, 116814);