#352994

order	512
nilpotency class	3
isoclinism	isoclinic class of #352174
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

1126 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1126 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)(7,13)(8,17)(9,11)(10,15)(12,14)(16,19)(18,20)(21,22)(23,29)(24,32)(25,27)(26,31)(28,33)(30,35)(34,36);(5,7,14,20,9,16,8,15)(6,10,17,19,11,18,12,13)(23,30)(26,34)(29,35)(31,36);(5,6)(7,10)(8,12)(9,11)(13,15)(14,17)(16,18)(19,20)(21,23,24,31,25,30,33,36)(22,26,28,29,27,34,32,35);(1,2,3,4)(6,11)(10,18)(12,17)(13,19)(22,27)(26,34)(28,32)(29,35)

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,c^b=ci,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, 260190);