C₂²×#1630 #6348022

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#1630

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3518 conjugacy classes of subgroups; only the 698 normal subgroups are shown.

normal subgroups only

This group has 3518 conjugacy classes of subgroups — too many to draw. Only its 698 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) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#18526, #72279943, C₂²×#18755, …
maximal normal subgroup of	C₂²×#18526, #72279943, C₂²×#18755, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255218);