#357370

order	512
nilpotency class	3
isoclinism	isoclinic class of #23360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 314 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) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=egi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,h^a=hi,c^b=cef,d^b=de,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264566);