C₂×#10338 #233537

order	512
nilpotency class	3
isoclinism	isoclinic class of #10336
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#10338

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3166 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 3166 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#233538, #72257491, #72257581, …
maximal normal subgroup of	C₂×#233538, #72257491, #72257581, …
maximal quotient of	#72257581, #72264375, #72264388, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140733);