C₂×#2659 #34572

order	256
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#2659

Subgroups

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

1491 conjugacy classes of subgroups; only the 607 normal subgroups are shown.

normal subgroups only

This group has 1491 conjugacy classes of subgroups — too many to draw. Only its 607 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) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#111268, #112271, #122017, …
maximal normal subgroup of	#111268, #112271, #122017, …
maximal quotient of	C₂×#14275, C₂×#14281, C₂×#14327, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27560);