C₂×#2614 #34598

order	256
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	Aut(G) order 12582912
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×#2614

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵
prime radical	C₂³×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

2597 conjugacy classes of subgroups; only the 607 normal subgroups are shown.

normal subgroups only

This group has 2597 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) = 26
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72268804
Frattini subgroup of	#72268802, #72268804, #72268809
Fitting subgroup of	C₂×#2614⋊C₃, #72227733
maximal subgroup of	#115180, #115634, #133206, …
maximal normal subgroup of	#115180, #115634, #133206, …
maximal quotient of	C₂×#14256, C₂×#14263, C₂×#14301, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27586);