#298305

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#1656	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

986 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=fh,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,g^c=gh

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, 205501);