#353027

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1205 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1205 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|b^2=eh,c^2=fi,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, 260223);