C₉×#555 #10589987

order	576
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₉×#555

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂⁴
commutator	C₂²	C₂⁴×C₉
Frattini, socle, cosocle	C₂²×C₃	C₂⁴×C₃
prime radical	C₃×#555	C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ G

354 conjugacy classes of subgroups; only the 225 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 225 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) = 25
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 4 generators on 25 points:

(1,2,3,4,5,6,7,8,9)(10,11)(12,16)(13,17)(14,15)(19,20)(23,24);(11,15)(13,16)(18,19,22,20)(21,24,25,23);(10,12,14,17)(11,16,15,13)(18,20,22,19)(21,24,25,23);(10,13,14,16)(11,17,15,12)(18,21,22,25)(19,23,20,24)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11779075, #11779485, #11780042, …
maximal subgroup of	C₉×#1624, C₉×#1634, C₉×#1646, …
maximal normal subgroup of	C₉×#1624, C₉×#1634, C₉×#1646, …
maximal quotient of	C₉×#2346, C₉×#2354, C₉×#2363, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 1869);