C₂×#13834 #513765

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁵×C₄³

Aliases

direct product

C₂×#13834

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4346 conjugacy classes of subgroups; only the 1858 normal subgroups are shown.

normal subgroups only

This group has 4346 conjugacy classes of subgroups — too many to draw. Only its 1858 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) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#13834⋊C₃
maximal subgroup of	C₂×#147497, #72266774, #72266899, …
maximal normal subgroup of	C₂×#147497, #72266774, #72266899, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420961);