C₂×#2621 #34626

order	256
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#2621

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

1523 conjugacy classes of subgroups; only the 567 normal subgroups are shown.

normal subgroups only

This group has 1523 conjugacy classes of subgroups — too many to draw. Only its 567 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);(7,8,9,10)(12,16)(17,19)(18,24)(25,26);(4,6)(11,12)(13,19)(14,18)(15,16)(17,21)(20,26)(22,25)(23,24);(3,4)(5,6)(7,8)(9,10)(11,13,15,21)(12,17,16,19)(14,22,23,20)(18,26,24,25);(11,14,15,23)(12,18,16,24)(13,20,21,22)(17,25,19,26)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#97040, #111398, #111630, …
maximal normal subgroup of	#97040, #111398, #111630, …
maximal quotient of	C₂×#14258, C₂×#14261, C₂×#14274, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27614);