C₉×#563 #10589995

order	576
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₉×#563

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂⁴
commutator	C₂²	C₂⁴×C₉
Frattini, socle, cosocle	C₂²×C₃	C₂⁴×C₃
prime radical	C₂×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ G

294 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 213 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) = 33
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 4 generators on 33 points:

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

Conjugacy classes & character table

171 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11780514, #11780594, #11780739, …
maximal subgroup of	C₉×#1657, C₉×#1660, C₉×#1664, …
maximal normal subgroup of	C₉×#1657, C₉×#1660, C₉×#1664, …
maximal quotient of	C₉×#2376, C₉×#2378, C₉×#2379, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 1877);