C₂²×#1709 #6357074

order	512
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₈

Aliases

direct product

C₂²×#1709

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂²×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2954 conjugacy classes of subgroups; only the 818 normal subgroups are shown.

normal subgroups only

This group has 2954 conjugacy classes of subgroups — too many to draw. Only its 818 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);(6,7)(8,9)(11,12)(13,14)(15,21)(16,24)(17,19)(18,22)(20,25)(23,27)(26,28);(5,6,9,11,10,12,8,7)(13,15,16,22,17,23,25,28)(14,18,20,21,19,26,24,27);(5,7,8,12,10,11,9,6)(14,19)(18,26)(20,24)(21,27)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264270);