C₂²×#1608 #6346342

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#1608

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁶	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3774 conjugacy classes of subgroups; only the 698 normal subgroups are shown.

normal subgroups only

This group has 3774 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) = 128

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253538);