C₂×#14170 #514637

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

direct product

C₂×#14170

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂⁴
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4802 conjugacy classes of subgroups; only the 1410 normal subgroups are shown.

normal subgroups only

This group has 4802 conjugacy classes of subgroups — too many to draw. Only its 1410 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);(4,6)(7,8,9,10)(19,20,23,26)(21,25,22,24);(11,12,14,16)(13,15,17,18)(19,21,23,22)(20,24,26,25);(3,4,5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,22,23,21)(20,25,26,24);(11,13,14,17)(12,15,16,18)(19,20)(21,24)(22,25)(23,26)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#754560, C₂×#581593, C₂×#4928842, …
maximal normal subgroup of	C₂×#754560, C₂×#581593, C₂×#4928842, …
maximal quotient of	C₂×#153867

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 421833);