C₂×#2646 #34564

order	256
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#2646

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

1187 conjugacy classes of subgroups; only the 527 normal subgroups are shown.

normal subgroups only

This group has 1187 conjugacy classes of subgroups — too many to draw. Only its 527 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) = 128

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#122018, C₂×#10344, C₂×#10345, …
maximal normal subgroup of	#122018, C₂×#10344, C₂×#10345, …
maximal quotient of	C₂×#14265, C₂×#14275, C₂×#14329, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27552);