#348310

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

850 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 160 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)(6,11)(7,14)(8,17)(10,19)(12,18)(15,16)(22,24)(23,29)(26,33)(27,35)(30,36)(31,34);(1,2,3,4)(5,6)(7,12)(8,15)(9,11)(10,13)(14,17)(16,18)(19,20)(21,22)(23,27)(24,25)(26,28)(29,36)(30,31)(32,33)(34,35);(5,7)(6,12)(8,11)(9,15)(10,16)(13,18)(14,19)(17,20)(21,23,25,31)(22,27,24,30)(26,34,33,29)(28,35,32,36);(5,8,9,12)(6,7,11,15)(10,18,19,17)(13,14,20,16)(21,24,25,22)(23,30,31,27)(26,28,33,32)(29,36,34,35)

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,d^b=dh,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255506);