C₉×#559 #10589991

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

Aliases

direct product

C₉×#559

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂⁴
commutator	C₂²	C₂⁴×C₉
Frattini, socle, cosocle	C₂²×C₃	C₂⁴×C₃
prime radical	C₃×#127	C₂×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)(10,11,14,13)(12,16,17,15)(19,22)(23,24);(11,13)(15,16)(18,19,21,22)(20,24,25,23);(10,12,14,17)(11,15,13,16);(10,13,14,11)(12,16,17,15)(18,20,21,25)(19,23,22,24)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11778917, #11779328, #11779757, …
maximal subgroup of	C₉×#1620, C₉×#1630, C₉×#1640, …
maximal normal subgroup of	C₉×#1620, C₉×#1630, C₉×#1640, …
maximal quotient of	C₉×#2355, C₉×#2357, C₉×#2366, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 1873);