#353014

order	512
nilpotency class	3
isoclinism	isoclinic class of #352064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1000 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:

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

Polycyclic presentation:

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

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, 260210);