C₂×#20019 #6358415

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈²
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂×#20019

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1682 conjugacy classes of subgroups; only the 530 normal subgroups are shown.

normal subgroups only

This group has 1682 conjugacy classes of subgroups — too many to draw. Only its 530 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) = 256

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

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265611);