C₂×#2097 #23729

order	256
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	Aut(G) order 262144
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#2097

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

575 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 215 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) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 30 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#2097, #359304, #359307, …
maximal normal subgroup of	C₄×#2097, #359304, #359307, …
maximal quotient of	C₂×#10664, #243849, #243863, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16717);