C₂×#19753 #6357774

order	512
nilpotency class	3
isoclinism	isoclinic class of #19745
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂×#19753

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2140 conjugacy classes of subgroups; only the 594 normal subgroups are shown.

normal subgroups only

This group has 2140 conjugacy classes of subgroups — too many to draw. Only its 594 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) = 34
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(3,4)(5,11)(6,15)(7,9)(8,14)(10,12)(13,17)(16,18)(19,20)(21,27)(22,24)(23,31)(25,29)(26,30)(28,33)(32,34);(3,5,12,18,7,13,6,14)(4,8,15,17,9,16,10,11)(20,24)(25,32)(26,31)(27,33);(4,9)(8,16)(10,15)(11,17)(19,21,23,29,22,28,30,34)(20,25,26,27,24,32,31,33);(5,13)(8,16)(11,17)(14,18)(21,28)(25,32)(27,33)(29,34)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264970);