C₂×#3028 #60397

order	256
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×#3028

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#120
commutator, Frattini, cosocle	C₂³	C₂⁵
prime radical	C₂×#585	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₂ ▷ C₁

2395 conjugacy classes of subgroups; only the 559 normal subgroups are shown.

normal subgroups only

This group has 2395 conjugacy classes of subgroups — too many to draw. Only its 559 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(3,4,8,12)(5,9,15,22)(6,10,17,24)(7,18,19,11)(13,25,27,20)(14,28,29,21)(16,30,31,23)(26,32,34,33);(3,5)(4,9)(6,13)(7,14)(8,15)(10,20)(11,21)(12,22)(16,26)(17,27)(18,28)(19,29)(23,32)(24,25)(30,33)(31,34);(3,6)(4,10)(5,13)(7,16)(8,17)(9,20)(11,23)(12,24)(14,26)(15,27)(18,30)(19,31)(21,32)(22,25)(28,33)(29,34);(3,7,8,19)(4,11,12,18)(5,14,15,29)(6,16,17,31)(9,21,22,28)(10,23,24,30)(13,26,27,34)(20,32,25,33)

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#3028⋊C₃, C₂×#11714176, C₂×#11714180, …
maximal subgroup of	#147754, #147763, #147765, …
maximal normal subgroup of	#147754, #147763, #147765, …
maximal quotient of	C₂×#18308, C₂×#18309, C₂×#18310, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53385);