C₂×#2817 #36649

order	256
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#2817

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

1239 conjugacy classes of subgroups; only the 487 normal subgroups are shown.

normal subgroups only

This group has 1239 conjugacy classes of subgroups — too many to draw. Only its 487 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) = 128

A minimal faithful permutation representation — 5 generators on 30 points:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#111324, #111934, #131305, …
maximal normal subgroup of	#111324, #111934, #131305, …
maximal quotient of	C₂×#14626, C₂×#14627, C₂×#14629, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29637);