#206226

order	512
nilpotency class	3
isoclinism	isoclinic class of #19651
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1124 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113422);