#353018

order	512
nilpotency class	3
isoclinism	isoclinic class of #352081
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)(7,15,17,18)(8,12,11,20)(9,16,19,13)(23,29)(24,31)(26,32)(28,34)(30,35)(33,36);(5,7,9,12,10,17,19,20)(6,11,13,15,14,8,16,18)(21,22)(23,26)(24,28)(25,27)(29,32)(30,33)(31,34)(35,36);(5,8)(6,12)(7,16)(9,18)(10,11)(13,17)(14,20)(15,19)(21,23,24,29,25,30,31,35)(22,26,28,32,27,33,34,36);(1,2,3,4)(7,17)(8,11)(12,20)(15,18)(22,27)(26,33)(28,34)(32,36)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,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=dhi,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, 260214);