#353008

order	512
nilpotency class	3
isoclinism	isoclinic class of #352064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²×C₈

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; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1000 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)(5,6)(7,14)(8,11)(9,17)(10,12)(13,18)(15,20)(16,19)(21,22)(23,28)(24,33)(25,35)(26,29)(27,32)(30,34)(31,36);(5,7,9,11,10,15,18,19)(6,8,13,14,12,16,17,20)(21,23,25,32,26,31,34,24)(22,27,30,28,29,33,35,36);(5,8,10,16)(6,11,12,19)(7,13,15,17)(9,14,18,20)(21,24,34,31,26,32,25,23)(22,28,35,27,29,36,30,33);(1,2,3,4)(6,12)(8,16)(13,17)(14,20)(22,29)(23,31)(24,32)(30,35)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,d^2=gh,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, 260204);