C₂×#19733 #6357375

order	512
nilpotency class	3
isoclinism	isoclinic class of #19733
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂×#19733

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

2068 conjugacy classes of subgroups; only the 594 normal subgroups are shown.

normal subgroups only

This group has 2068 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);(4,8)(6,13)(9,14)(10,15)(12,17)(16,18)(21,27)(22,29)(24,30)(26,32)(28,33)(31,34);(3,4,6,8,7,10,13,15)(5,9,12,14,11,16,17,18)(19,20)(21,24)(22,26)(23,25)(27,30)(28,31)(29,32)(33,34);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,21,22,27,23,28,29,33)(20,24,26,30,25,31,32,34);(4,10)(5,11)(8,15)(12,17)(20,25)(24,31)(26,32)(30,34)

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, 6264571);